Contents

Chapter 5. An analytic approach to SPDEs 185 1. Introduction 185 2. Generalities 186 3. The Stochastic Banach Spaces 190 4. Model Equations 195 5. Equations with Variable Coefficients 206 6. Proof of Theorem 5.1 213 7. Embedding Theorems for H_{p}^{n}(τ) 218 8. Applications 223 9. Open Problems 237

Bibliography 239

Index 241

CHAPTER 5 An analytic approach to SPDEs

N.V. KRYLOV 127 Vincent Hall

University of Minnesota Minneapolis, MN, 55455

                                                                           krylov@math.umn.edu

1. Introduction

Evolutional stochastic partial differential equations (SPDEs) arise in many applications of probability theory and have been treated since long ago (see [30]). An example of a linear second-order SPDE is given by the following equation in ℝ^{d}:

du = (a^{ij}u_{x^{i}x^{j}} + b^{i}u_{x^{i}} + cu + f) dt + (σ^{ik}u_{x^{i}} + ν^{k}u + g^{k}) dw_{t}^{k}, t > 0,  (1.1)

where summation with respect to the repeated indices i, j, k is assumed as usual, i and j go from 1 to d, and k may run through 1, 2, ....

The main purpose of this publication is to present a theory of solvability of the Cauchy problem for linear and some quasi-linear equations like (1.1) in spaces of summable functions with exponent of summability p ≥ 2. If p = 2, so that we are concerned with solutions belonging to the Sobolev spaces W_{2}^{n}(ℝ^{d}), such a theory does exist and is rather complete and satisfactory (see, for instance, [30]). Recall that W_{2}^{n}(ℝ^{d}) is the set of all generalized functions on ℝ^{d} whose derivatives up to and including the nth order belong to L_{2}(ℝ^{d}). Some results concerning the solvability of the first boundary-value problem in spaces like W_{2}^{n}(D), where D is a smooth domain, can be found in [2], [6], [20], and [32]. Roughly speaking, the main tool in W_{2}^{n}-theory is integration by parts. There are also approaches based on semigroup methods [5], [6], which work well for the equations with nonrandom leading coefficients a^{ij} and again in the Hilbert-space framework.

The necessity of the L_{p}–theory arises, for instance, when one wants to find the solutions numerically. The convergence rate and the way the finite difference should be chosen depend on smoothness properties of solutions. Here by the smoothness properties we mean continuity, Hölder continuity, differentiability, continuity of derivatives, and so on.

One of inconveniences of W_{2}^{n}-theory is that W_{2}^{n}(ℝ^{d}) ⊂ C^{n−d/2}(ℝ^{d}) only if 2n > d, and one can prove that the solutions belong to W_{2}^{n}(ℝ^{d}) only if the coefficients are n − 2 times continuously differentiable. Therefore, if we want to get the solutions m times continuously differentiable with respect to x ∈ ℝ^{d}, we have to suppose that the coefficients of the equation are more than m+d/2−2 times continuously differentiable even if the free terms are of class C_{0}^{∞}(ℝ^{d}). At the same time, W_{p}^{n}(ℝ^{d}) ⊂ C^{n−d/p}(ℝ^{d}) if pn > d, and, by taking p sufficiently large, we see that the solutions have almost as many usual derivatives as generalized ones. Actually, exactly for this purpose the spaces W_{p}^{n}(ℝ^{d}) with p ≥ 2 have already been used in SPDE theory (see, for instance, [30]), but the corresponding results, obtained again by integration by parts, were not sharp. It is worth mentioning that sharp results concerning C^{2+α}(ℝ^{d})–theory are recently obtained in [27] for equations like (1.1) but with σ ≡ 0.

Another advantage of the W_{p}^{n} setting with p ≥ 2 can be seen in the case of very popular equations with so-called cylindrical white noise (see, for instance, [8], [26], [28], [35], and references therein). Although these equations are covered by the general W_{p}^{n}-theory for any p ≥ 2 (see Section 8.3), for p = 2 we get only the solutions summable to any degree, and the solutions become continuous only for p > 2. By the way, as in [30], we consider n positive and negative, but in contrast with [30] we allow n to have non-integer values. For general n we are working in the spaces of Bessel potentials H_{p}^{n}(ℝ^{d}), and, in the case of equations with cylindrical white noise, we take n slightly less than (−3/2).

Our main tool is the theory of spaces H_{p}^{n}(ℝ^{d}), borrowed from [33], together with a result from [17] or [18] which is an analog of the so-called maximal regularity property of stochastic convolutions in Hilbert spaces obtained by Da Prato (cf. [4]). We also use some results from the theory of parabolic equations and follow a general scheme of proving the solvability of PDEs adopted in this theory. We discuss this general scheme in Sec. 2.

The main source of these notes is author’s article [20]. However, it underwent a major revision and restructuring here, so that, in author’s opinion, the whole subject looks more natural now. Also, the proofs are given with much more details, and several mistakes have been corrected. In this regard the contribution of S. Lototsky, H. Yoo, and A. Zatezalo is greatly appreciated. The author has delivered four lectures on the subject at Workshop/School on SPDEs, Theory and Applications, Los Angeles, January, 3–7, 1996. This was a very good opportunity to think all over again. It is a pleasure to thank the organizers of the Workshop for the invitation and hospitality.

2. Generalities

We want to explain here some basic ideas guiding the investigation we are going to present further.

Rewrite equation (1.1) in the following form

du = (Lu + f) dt + (Λ^{k}u + g^{k}) dw_{t}^{k}, t > 0,  (2.1)

where

Lu = a^{ij}u_{x^{i}x^{j}} + b^{i}u_{x^{i}} + cu, Λ^{k}u = σ^{ik}u_{x^{i}} + ν^{k}u.

Recall that we are using the summation convention. Of course, w_{t}^{k} are independent Wiener processes and du is Itô’s stochastic differential with respect to t. The equation is considered for all x ∈ ℝ^{d} and t > 0. The coefficients σ, ν, and the “free” term g may vanish, and then (2.1) becomes a usual parabolic partial differential equation. Therefore, it is natural to recall some ideas and results from the L_{p}-theory of parabolic partial differential equations.

Let ℝ^{d} be a d-dimensional Euclidean space of points x = (x^{1}, ..., x^{d}). By a distribution or a generalized function on ℝ^{d} we mean an element of the space D of real-valued Schwarz distributions defined on C_{0}^{∞}, where C_{0}^{∞} = C_{0}^{∞}(ℝ^{d}) is the set of all infinitely differentiable functions with compact support.

For given p ∈ (1, ∞) and n ∈ (−∞, ∞), define the space H_{p}^{n} = H_{p}^{n}(ℝ^{d}) (called the space of Bessel potentials or the Sobolev space with fractional derivatives) as the space of all generalized functions u such that (1 − Δ)^{n/2}u ∈ L_{p} = L_{p}(ℝ^{d}). To explain the meaning of this, let us first introduce (1 − Δ)^{n/2} in the following way (see [14]). If α ∈ (0, 1), then, for a constant c(α) and all z ≤ 0,

(1 − z)^{α} = c(α) ∫_{0}^{∞} \frac{e^{−t}e^{zt} − 1}{t^{α}} \frac{dt}{t}.

If we formally substitute here Δ instead of z, then we get the following definition of (1−Δ)^{α}

(1 − Δ)^{α}u = c(α) ∫_{0}^{∞} \frac{e^{−t}T_{t}u − u}{t^{α}} \frac{dt}{t},  (2.2)

where T_{t} is the semigroup associated with Δ, that is, a solution of T′_{t} = ΔT_{t} similarly to e^{zt} which is a solution of f′ = zf. The following is an explicit formula for T_{t}:

T_{t}u(x) := \frac{1}{(4πt)^{d/2}} ∫_{ℝ^{d}} u(y)e^{− \frac{1}{4t} |x−y|^{2}} dy.  (2.3)

In the same way for any α > 0 we define

(1 − Δ)^{−α}u = d(α) ∫_{0}^{∞} t^{α}e^{−t}T_{t}u \frac{dt}{t},  (2.4)

where d(α) is an appropriate constant. It turns out (see [14]) that formulas (2.2) and (2.4) are sufficient to consistently define (1−Δ)^{n/2} for any n ∈ (−∞, ∞). The result of application of (1 − Δ)^{−n/2} to an f ∈ L_{p} is defined as a limit of truncated integrals in (2.2) or (2.4). For a distribution u, we say that u ∈ H_{p}^{n} if there is f ∈ L_{p} such that u = (1 − Δ)^{−n/2}f in the sense of distributions, that is, if u and (1 − Δ)^{−n/2}f coincide as elements in the space D. In this case we also write (1 − Δ)^{n/2}u = f.

For u ∈ H_{p}^{n} one introduces the norm

||u||_{n,p} := ||(1 − Δ)^{n/2}u||_{p},

where ||·||_{p} is the norm in L_{p}. It is known (see, for instance, [33], [34]) that H_{p}^{n} is a Banach space with norm || · ||_{n,p} and the set C_{0}^{∞} is dense in H_{p}^{n}.

Next, for fixed T one introduces the space H_{p}^{1,2}(T) = H_{p}^{1,2}((0, T) × ℝ^{d}) as

\{u = u(t, x) : ||u||_{1,2,p}^{p} := ∫_{0}^{T} ||\frac{∂u}{∂t}(t, ·)||_{p}^{p} dt + ∫_{0}^{T} ||u(t, ·)||_{2,p}^{p} dt < ∞\}.

The norm || · ||_{1,2,p} makes H_{p}^{1,2}(T) a Banach space.

The investigation of the deterministic counterpart of equation (2.1)

\frac{∂u}{∂t} = Lu + f  (2.5)

with zero initial condition goes in the following way (see [31]). First, for the simplest equation

\frac{∂u}{∂t} = Δu + f,  (2.6)

its solvability in H_{p}^{1,2}(T) is proved by means of explicit formulas and some estimates of heat potentials, provided that f ∈ L_{p}((0, T) × ℝ^{d}). In Subsec. 4.1 we use the following theorem which is proved in [31] (also see Remark 2.3.2 in [34]).

Theorem 2.1. For any f ∈ L_{p}((0, T) × ℝ^{d}) and u_{0} ∈ H_{p}^{2−2/p} there exists a unique solution u ∈ H_{p}^{1,2}(T) of the heat equation (2.6) with initial data u(0) = u_{0}. In addition,

||u_{xx}||_{L_{p}((0,T)×ℝ^{d})} + ||\frac{∂u}{∂t}||_{L_{p}((0,T)×ℝ^{d})} ≤ N(d, p)(||f||_{L_{p}((0,T)×ℝ^{d})} + ||u_{0}||_{2−2/p,p}), (2.7)
||u||_{1,2,p} ≤ N(d, p, T)(||f||_{L_{p}((0,T)×ℝ^{d})} + ||u_{0}||_{2−2/p,p}),

where u_{xx} is the matrix of second–order derivatives of u with respect to x.

This theorem yields a bounded operator R_{1} which maps any f ∈ L_{p}((0, T) × ℝ^{d}) into the solution u ∈ H_{p}^{1,2}(T) of the heat equation (2.6) with zero initial data.

Then, the so–called a priori estimate is obtained for (2.5). One assumes that there is a solution u ∈ H_{p}^{1,2}(T) of (2.5) with zero initial condition and inequality (2.7) is proved, where N is a constant probably depending on T and some characteristics of L (below, N, usually without indices or arguments, denotes various constants, and writing N(d, p) is just to indicate that N depends only on d and p). At this point, one of the most fruitful general ideas in the theory of partial differential equations, linear or not, is used. The idea says that obtaining a priori estimates for solutions in a class of functions implies that the equation is solvable in the same class. We will see how this works below.

Of course, this idea only works “as usual”. For instance, there is no trouble to estimate the sup norm of the second order derivative for possible solutions of the equation (u′′)^{2} = f on [0, 1] satisfying u(0) = 0 and u(1) = 1, where f is a given bounded function. So, one has an a priori estimate, but nevertheless the set of solutions is empty if f < 0. Also, the statement “if (2.5) has a solution u ∈ H_{p}^{1,2}(T), then (2.7) holds” looks bad from the point of view of formal logic (in which any statement with a false assumption is true). In fact, instead of (2.7), one proves

||u||_{1,2,p} ≤ N||Lu − \frac{∂u}{∂t}||_{L_{p}((0,T)×ℝ^{d})}  (2.8)

for any u ∈ H_{p}^{1,2}(T) such that in a certain sense u(0, ·) = 0. A usual way to prove (2.8) consists of observing that (2.8) is true for L = Δ, then applying perturbation methods to get (2.8) for L that are close enough to Δ, and finally, replacing Δ with other operators with constant coefficients and using partitions of unity in order to get small regions in which L is close to an operator with constant coefficients (we will see this in details in the proof of Theorem 5.1).

The last step is to use the method of continuity. Instead of (2.5) one considers the following family of equations

\frac{∂u}{∂t} = L_{λ}u + f  (2.9)

with λ ∈ [0, 1], where

L_{λ}u = λΔ + (1 − λ)L.

Assume that the a priori estimate (2.8) holds with the same constant N for all L_{λ} in place of L. Also assume that, for a λ = λ_{0} ∈ [0, 1], equation (2.9) with zero initial data has a unique solution u ∈ H_{p}^{1,2}(T) for any f ∈ L_{p}((0, T) × ℝ^{d}). Then we have an operator R_{λ_{0}} such that R_{λ_{0}}f = u. By the way, this assumption is satisfied for λ_{0} = 1 by Theorem 2.1. From (2.8) we get that

||R_{λ_{0}}f||_{1,2,p} ≤ N||f||_{L_{p}((0,T)×ℝ^{d})}.  (2.10)

For other λ ∈ [0, 1] we rewrite (2.9) as

\frac{∂u}{∂t} = L_{λ_{0}}u + \{(λ − λ_{0})(Δ − L)u + f\}, u = R_{λ_{0}}\{(λ − λ_{0})(Δ − L)u + f\}

and we solve the last equation by iterations. Define u_{0} = 0 and

u_{n+1} = R_{λ_{0}}{(λ − λ_{0})(Δ − L)u_{n} + f}.

Then by (2.10)

||u_{n+1} − u_{n}||_{1,2,p} ≤ N|λ − λ_{0}| ||(Δ − L)(u_{n} − u_{n−1})||_{L_{p}((0,T)×ℝ^{d})}

≤ N_{1}|λ − λ_{0}| ||u_{n} − u_{n−1}||_{1,2,p},

and we have a strongly convergent sequence u_{n} in H_{p}^{1,2}(T) if N_{1}|λ−λ_{0}| ≤ 1/2. Consequently, one can solve (2.9) for such λ. Starting from λ = 1, one reaches λ = 0 in finitely many steps, and this finishes the proof of solvability of (2.5).

There are two central objects in the above argument. These are the Banach space H_{p}^{1,2}(T) and the operator L−∂/∂t : H_{p}^{1,2}(T) → L_{p}((0, T)×ℝ^{d}). Since we want to implement the same kind of argument for equations like (2.1), the first thing to do is to find an appropriate counterpart of H_{p}^{1,2}(T). This was a problem for some time, since one cannot expect any differentiability property with respect to t for solutions u of (2.1). Then an observation appeared that H_{p}^{1,2}(T) can also be defined without using ∂u/∂t. Indeed, almost obviously,

H_{p}^{1,2}(T) = {u : u(t, x) = u(0, x) + ∫_{0}^{t} f(s, x) ds, u, u_{x}, u_{xx}, f ∈ L_{p}((0, T) × ℝ^{d})}.

Somewhat unusual about this definition is that in the formula

u(t, x) = u(0, x) + ∫_{0}^{t} f(s, x) ds

the function f only belongs to L_{p}((0, T) × ℝ^{d}) but u, u_{x}, u_{xx} also belong to L_{p}((0, T) × ℝ^{d}) (by definition).

Now the guess is natural that a stochastic counterpart H_{p}^{2}(T) of the spaces H_{p}^{1,2}(T) could be the space of functions u = u(ω, t, x) such that

u(t, x) = u(0, x) + ∫_{0}^{t} f(s, x) ds + ∫_{0}^{t} g^{k}(s, x) dw_{s}^{k},  (2.11)

E ∫_{0}^{T} ∫_{ℝ^{d}}{|u| + |u_{x}| + |u_{xx}| + |f|}^{p} dxdt < ∞,

and something of the same type is satisfied for g = {g^{k}}. It may look a little bit surprising that one needs p ≥ 2 and

E ∫_{0}^{T} ∫_{ℝ^{d}}{|g| + |g_{x}|}^{p} dxdt < ∞,

which involves both g and g_{x}, where

|g|^{2} := ∑_{k=1}^{∞} |g^{k}|^{2}, |g_{x}|^{2} := ∑_{k=1}^{∞} |g_{x}^{k}|^{2}.

We will explain later why one needs p ≥ 2 (see Remark 4.3). The need for some conditions on g_{x} may be explained by the fact that in parabolic equations one derivative in t is worth two derivatives in x. The stochastic integral in (2.11) has, so to speak, one half derivative in time, so one can expect that it also has one derivative in x. The second derivative should be provided by differentiability of g itself. This is a kind of phenomenological explanation. The real explanation, however, is that with such definition of stochastic H_{p}^{2}(T) spaces one can construct a theory completely analogous to the theory discussed above.

For p = 2 we can also see the necessity of imposing conditions on g_{x} from the model one–dimensional equation

du = u_{xx} dt + g dw_{t},

where x ∈ ℝ and w_{t} is a one–dimensional Wiener process. Forgetting for a while about rigor, by Itô’s formula we get

d(u^{2}) = (2uu_{xx} + g^{2}) dt + 2ug dw_{t}, ∫_{ℝ} u^{2}(t, x) dx = ∫_{ℝ} u^{2}(0, x) dx

+2 ∫_{0}^{t} ∫_{ℝ} uu_{xx}(s, x) dx ds + ∫_{0}^{t} ∫_{ℝ} g^{2}(s, x) dx ds + ∫_{0}^{t} ∫_{ℝ} ug(s, x) dx dw_{s}.

Integrating by parts we see that

E||u(t, ·)||_{2}^{2} + 2E ∫_{0}^{t} ||u_{x}(s, ·)||_{2}^{2} ds = E||u(0, ·)||_{2}^{2} + E ∫_{0}^{t} ||g(s, ·)||_{2}^{2} ds.

This shows that without assumptions on g_{x} we only get an estimate of the first derivative of u.

With the spaces H_{p}^{2}(T) at hand, we write (2.1) in an operator form by introducing the operator (L, Λ) which can be applied to any element u ∈ H_{p}^{2}(T). Namely for a u ∈ H_{p}^{2}(T) we write (L, Λ)u = −(f, g) if and only if

u(t) = u(0) + ∫_{0}^{t}[Lu + f](s) ds + ∫_{0}^{t}[Λ^{k}u + g^{k}](s) dw_{s}^{k}.

The expression (L, Λ)u does make sense for any u ∈ H_{p}^{2}(T) since, if we have (2.11), then (L, Λ)u = (Lu − f, Λu − g). Now, instead of equation (2.1) we have the equation (L, Λ)u = −(f, g) in the Banach space H_{p}^{2}(T), which makes the situation completely analogous to the one in the theory of parabolic equations.

This finishes our explanation of the basics of the theory, and now we are ready to present the details.

3. The Stochastic Banach Spaces

Let (Ω, F, P) be a complete probability space, (F_{t}, t ≥ 0) be an increasing filtration of σ-fields F_{t} ⊂ F containing all P -null subsets of Ω, and P be the predictable σ-field generated by (F_{t}, t ≥ 0). Let {w_{t}^{k}; k = 1, 2, ...} be a family of independent one–dimensional F_{t}–adapted Wiener processes defined on (Ω, F, P).

We fix a p ≥ 2 and an integer d ≥ 1 and use the notation C_{0}^{∞}, D, L_{p}, H_{p}^{n}, || · ||_{p}, and || · ||_{n,p} introduced in Sec. 2. We are dealing with distributions, and, for a distribution u and ϕ ∈ C_{0}^{∞}, by (u, ϕ) or (ϕ, u) we mean the result of application of u to ϕ. Observe that, for u ∈ H_{p}^{n} and ϕ ∈ C_{0}^{∞}, by definition

(u, ϕ) = ((1 − Δ)^{n/2}u, (1 − Δ)^{−n/2}ϕ) = ∫_{ℝ^{d}}[(1 − Δ)^{n/2}u](x)(1 − Δ)^{−n/2}ϕ(x) dx, (3.1)

where the last integral is a usual Lebesgue integral. Since (1 − Δ)^{n/2}u ∈ L_{p}, one can define (u, ϕ) by (3.1) for any ϕ whose derivatives vanish sufficiently fast at infinity, say exponentially fast.

Sometimes it is useful to notice that || · ||_{n,p} ≤ || · ||_{m,p} for m ≥ n. Recall that the set C_{0}^{∞} is dense in H_{p}^{n} and the latter is a subset of D (see, for instance, Theorem 2.3 (ii) of [33]). Also recall (see, for instance [33]) that, for integers n ≥ 0, the space H_{p}^{n} coincides with the Sobolev space W_{p}^{n} = W_{p}^{n}(ℝ^{d}).

We apply the same definitions to l_{2}-valued functions g, where l_{2} is the set of all realvalued sequences g = {g^{k}; k = 1, 2, ...} with the norm defined by |g|_{l_{2}}^{2} := ∑_{k} |g^{k}|^{2}. Specifically,

||g||_{p} := || |g|_{l_{2}}||_{p}, ||g||_{n,p} := || |(1 − Δ)^{n/2}g|_{l_{2}}||_{p}.

Finally, for stopping times τ, we denote ^{|}(0, τ]] = {(ω, t) : 0 < t ≤ τ(ω)}

ℍ_{p}^{n}(τ) = L_{p}(^{|}(0, τ]], P, H_{p}^{n}), ℍ_{p}^{n} = ℍ_{p}^{n}(∞),
ℍ_{p}^{n}(τ, l_{2}) = L_{p}(^{|}(0, τ]], P, H_{p}^{n}(ℝ^{d}, l_{2})), 𝕃_{...}... = ℍ_{...}^{0}... .

The norms in these spaces are defined in an obvious way. By convention, elements of spaces like ℍ_{p}^{n} are treated as functions rather than distributions or classes of equivalent functions, and if we know that a function of this class has a modification with better properties, then we always consider this modification. For instance, if we take u ∈ H_{p}^{n} and n − d/p > 0, then u has a bounded continuous modification, but we talk about sup_{x} u(x) instead of sup of this modification. Also, elements of spaces ℍ_{p}^{n}(τ, l_{2}) need not be defined or belong to H_{p}^{n} for all (ω, t) ∈ ^{|}(0, τ]]. As usual, these properties are needed only for almost all (ω, t).

For n ∈ ℝ and

(f, g) ∈ F_{p}^{n}(τ) := ℍ_{p}^{n}(τ) × ℍ_{p}^{n+1}(τ, l_{2}),

set

||(f, g)||_{F_{p}^{n}(τ)} := ||f||_{ℍ_{p}^{n}(τ)} + ||g||_{ℍ_{p}^{n+1}(τ,l_{2})}.

Definition 3.1. For a D-valued function u ∈ ⋂_{T >0} ℍ_{p}^{n}(τ ∧ T), we write u ∈ H_{p}^{n}(τ) if u_{xx} ∈ ℍ_{p}^{n−2}(τ), u(0, ·) ∈ L_{p}(Ω, F_{0}, H_{p}^{n−2/p}), and there exists (f, g) ∈ F_{p}^{n−2}(τ) such that, for any ϕ ∈ C_{0}^{∞}, the equality

(u(t, ·), ϕ) = (u(0, ·), ϕ) + ∫_{0}^{t}(f(s, ·), ϕ) ds + ∑_{k=1}^{∞} ∫_{0}^{t}(g^{k}(s, ·), ϕ) dw_{s}^{k}  (3.2)

holds for all t ≤ τ with probability 1. We also define H_{p,0}^{n}(τ) = H_{p}^{n}(τ) ∩ {u : u(0, ·) = 0},

||u||_{H_{p}^{n}(τ)} = ||u_{xx}||_{ℍ_{p}^{n−2}(τ)} + ||(f, g)||_{F_{p}^{n−2}(τ)} + (E||u(0, ·)||_{n−2/p,p}^{p})^{1/p}.  (3.3)

As always, we drop τ in H_{p}^{n}(τ) and F_{p}^{n}(τ) if τ = ∞.

Remark 3.2. It is worth noting that the elements of H_{p}^{n}(τ) are assumed to be defined for all (ω, t) and take values in D. Obviously, H_{p}^{n}(τ) is a linear space. As usual, we identify two elements u_{1} and u_{2} of H_{p}^{n}(τ) if ||u_{1} −u_{2}||_{H_{p}^{n}(τ)} = 0. Actually, as we will see later, in this way we only identify indistinguishable functions. Also, observe that the series of stochastic integrals in (3.2) converges uniformly in t in probability on [0, τ ∧ T] for any finite T, since the quadratic variations of these stochastic integrals satisfy (cf. (3.1))

∑_{k=1}^{∞} ∫_{0}^{τ∧T} (g^{k}(s, ·), ϕ)^{2} ds = ∑_{k=1}^{∞} ∫_{0}^{τ∧T} ((1 − Δ)^{(n−1)/2}g^{k}(s, ·), (1 − Δ)^{(1−n)/2}ϕ)_{L_{2}}^{2} ds

≤ ||(1 − Δ)^{(1−n)/2}ϕ||_{1} ∫_{0}^{τ∧T} ∑_{k=1}^{∞}(|(1 − Δ)^{(n−1)/2}g^{k}(s, ·)|^{2}, |(1 − Δ)^{(1−n)/2}ϕ|)_{L_{2}} ds

≤ N ∫_{0}^{τ∧T} ||(∑_{k=1}^{∞} |(1 − Δ)^{(n−1)/2}g^{k}(s, ·)|^{2})^{1/2}||_{p}^{2} ds < ∞ (a.s.),

where N = ||(1 − Δ)^{(1−n)/2}ϕ||_{1} ||(1 − Δ)^{(1−n)/2}ϕ||_{q}, q = p/(p − 2) and we have used that p ≥ 2.

As a consequence of the uniform convergence, (u(t, ·), ϕ) is continuous in t on [0, τ ∧ T] for any finite T (a.s.).

Remark 3.3. There can exist only one couple (f, g) for which (3.2) holds. Indeed, if there are two, then one can represent zero as a sum of a continuous process of bounded variation and a continuous local martingale. One knows that this is only possible if both processes vanish. Therefore, the couple (f, g) is uniquely determined by u, and notation ||u||_{H_{p}^{n}(τ)} in (3.3) makes sense.

Remark 3.4. It is known that the operator (1 − Δ)^{m/2} maps isometrically H_{p}^{n} onto H_{p}^{n−m} for any n, m. Also, the inequalities from Remark 3.2 can be used to show that given u ∈ H_{p}^{n}(τ), one can in (3.2) take any infinitely differentiable function ϕ whose derivatives vanish sufficiently fast at infinity, say exponentially fast. This allows us to substitute (1 − Δ)^{m/2}ϕ in (3.2) instead of ϕ and shows that the operator (1 − Δ)^{m/2} maps isometrically H_{p}^{n}(τ) onto H_{p}^{n−m}(τ) for any n, m. The same is true for ℍ_{p}^{n}(τ).

Definition 3.5. For u ∈ H_{p}^{n}(τ), if (3.2) holds, then we write f = 𝔻u, g = 𝕊u (for “deterministic” and “stochastic” parts of u) and we also write

u(t) = u(0) + ∫_{0}^{t} 𝔻u(s) ds + ∫_{0}^{t} 𝕊^{k}u(s) dw_{s}^{k}, du = f dt + g^{k} dw_{t}^{k} t ≤ τ.

Remark 3.6. It follows from Definitions 3.1 and 3.5 that the operators 𝔻 and 𝕊 are continuous operators from H_{p}^{n}(τ) to ℍ_{p}^{n−2}(τ) and ℍ_{p}^{n−1}(τ, l_{2}) respectively. From Theorem 4.2 and Remark 3.4 it follows that 𝕊 maps H_{p}^{n}(τ) onto ℍ_{p}^{n−1}(τ, l_{2}). However, at this point we do not know how rich H_{p}^{n}(τ) is. Nevertheless obviously H_{p}^{1,2}(T) ⊂ H_{p}^{2}(T).

Theorem 3.7. The spaces H_{p}^{n}(τ) and H_{p,0}^{n}(τ) are Banach spaces with norm (3.3). In addition if τ ≤ T, where T is a finite constant, then for u ∈ H_{p}^{n}(τ)

||u||_{ℍ_{p}^{n}(τ)} ≤ N(d, T)||u||_{H_{p}^{n}(τ)}, E sup_{t≤τ} ||u(t, ·)||_{n−2,p}^{p} ≤ N(d, T)||u||_{H_{p}^{n}(τ)}^{p}.  (3.4)

Proof. We first deal with (3.4). Obviously

||u||_{ℍ_{p}^{n}(τ)} = ||(1 − Δ)u||_{ℍ_{p}^{n−2}(τ)} ≤ ||u||_{ℍ_{p}^{n−2}(τ)} + ||u||_{H_{p}^{n}(τ)},

so that to prove (3.4) we only need to prove that

E sup_{t≤τ} ||u(t, ·)||_{n−2,p}^{p} ≤ N||u||_{H_{p}^{n}(τ)}^{p}.  (3.5)

Owing to Remark 3.4 we may and will assume that n = 2. Take a nonnegative function ζ ∈ C_{0}^{∞} with unit integral, for ε > 0 define ζ_{ε}(x) = ε^{−d}ζ(x/ε), and for generalized functions u let u^{(ε)}(x) = u ∗ ζ_{ε}(x). Observe that u^{(ε)}(x) is a continuous (infinitely differentiable) function of x for any distribution u. Plugging in ζ_{ε}(· − x) instead of ϕ in (3.2), we get that for any x the equality

u^{(ε)}(t, x) = u^{(ε)}(0, x) + ∫_{0}^{t} f^{(ε)}(s, x) ds + ∑_{k=1}^{∞} ∫_{0}^{t} g^{(ε)k}(s, x) dw_{s}^{k}  (3.6)

holds almost surely for all t ≤ τ. If necessary, we redefine the stochastic integrals in (3.6) in such a way that (3.6) would hold for all ω, t, and x such that t ≤ τ. Here

E||u^{(ε)}(0, ·)||_{p}^{p} ≤ E||u(0, ·)||_{p}^{p} ≤ E||u(0, ·)||_{n−2/p,p}^{p} ≤ ||u||_{H_{p}^{n}(τ)}^{p},

where we use that, by Minkowski’s inequality, ||h^{(ε)}||_{p} ≤ ||ζ_{ε}||_{1} ||h||_{p} = ||h||_{p}. Similarly,

| ∫_{0}^{t} f^{(ε)}(s, x) ds|^{p} ≤ T^{p−1} ∫_{0}^{τ} |f^{(ε)}(s, x)|^{p} ds,

E sup_{t≤τ} || ∫_{0}^{t} f^{(ε)}(s, ·) ds||_{p}^{p} ≤ T^{p−1}E ∫_{0}^{τ} ||f(s, ·)||_{p}^{p} ds ≤ T^{p−1}||u||_{H_{p}^{n}(τ)}^{p}.

Finally, by Burkholder–Davis–Gundy inequalities

E sup_{t≤τ} | ∑_{k=1}^{∞} ∫_{0}^{t} g^{(ε)k}(s, x) dw_{s}^{k}|^{p} ≤ NE| ∫_{0}^{τ} ∑_{k=1}^{∞} |g^{(ε)k}(s, x)|^{2} ds|^{p/2}

= NE| ∫_{0}^{τ} |g^{(ε)}|_{l_{2}}^{2}(s, x) ds|^{p/2},

and as above (the first term below makes sense by virtue of (3.6))

E sup_{t≤τ} || ∑_{k=1}^{∞} ∫_{0}^{t} g^{(ε)k}(s, ·) dw_{s}^{k}||_{p}^{p} ≤ ∫_{ℝ^{d}} E sup_{t≤τ} | ∑_{k=1}^{∞} ∫_{0}^{t} g^{(ε)k}(s, x) dw_{s}^{k}|^{p} dx

≤ NE ∫_{ℝ^{d}} | ∫_{0}^{τ} |g^{(ε)}|_{l_{2}}^{2}(s, x) ds|^{p/2} dx ≤ NE(∫_{0}^{τ} || |g^{(ε)}|_{l_{2}}^{2}(s, ·)||_{p/2} ds)^{p/2}

= NE(∫_{0}^{τ} || |g^{(ε)}|_{l_{2}}(s, ·)||_{p}^{2} ds)^{p/2} ≤ NE ∫_{0}^{τ} || |g^{(ε)}|_{l_{2}}(s, ·)||_{p}^{p} ds
≤ N||g||_{𝕃_{p}(τ,l_{2})}^{p} ≤ N||u||_{H_{p}^{n}(τ)}^{p}.

This, along with (3.6), leads to

E sup_{t≤τ} ||u^{(ε)}(t, ·)||_{p}^{p} ≤ N||u||_{H_{p}^{n}(τ)}^{p}.  (3.7)

Furthermore, by using the fact that ||h^{(ε)} − h^{(γ)}||_{p} → 0 whenever h ∈ L_{p} and ε, γ → 0 and by considering u^{(1/m)} − u^{(1/k)} instead of u^{(ε)}, we easily see that that u^{(1/m)}(t ∧ τ, x) is a Cauchy sequence in L_{p}(Ω, B([0, T], L_{p})). Define ū as its limit in this space. Then, for a subsequence m′, we have u^{(1/m′)}(t, ·) → ū(t, ·) in L_{p} if t ≤ τ with probability 1. On the other, hand u^{(1/m)}(t, ·) → u(t, ·) in the sense of distributions for all ω and t such that t ≤ τ(ω). Therefore with probability one we have u(t, ·) ∈ L_{p} for t ≤ τ. Now, (3.7) and Fatou’s lemma yield (3.5) for n = 2. As explained above, this proves (3.4).

Next, we derive the first assertion of our theorem from (3.4). As usual, we have only to check the completeness of H_{p}^{n}(τ). If {u_{j}} is a Cauchy sequence in H_{p}^{n}(τ), then it is a Cauchy sequence in ℍ_{p}^{n}(τ ∧ T) for any T, and there is u ∈ ⋂_{T >0} ℍ_{p}^{n}(τ ∧ T) such that ||u − u_{j}||_{ℍ_{p}^{n}(τ∧T)} → 0 as j → ∞. Furthermore, u_{jxx} form a Cauchy sequence and therefore converge in ℍ_{p}^{n−2}(τ). It follows easily that ||u_{xx} − u_{jxx}||_{ℍ_{p}^{n−2}(τ)} → 0.

Also, for u_{j}(0), f_{j}, g_{j} corresponding to u_{j}, there is u(0) ∈ L_{p}(Ω, F_{0}, H_{p}^{n−2/p}) and (f, g) ∈ F_{p}^{n−2}(τ) such that

E||u(0) − u_{j}(0)||_{n−2/p,p}^{p} → 0, ||f − f_{j}||_{ℍ_{p}^{n−2}(τ)} → 0, ||g − g_{j}||_{ℍ_{p}^{n−1}(τ,l_{2})} → 0.

By using the argument from Remark 3.2, one can show that for any ϕ ∈ C_{0}^{∞} equality (3.2) holds in ^{|}(0, τ]] almost everywhere.

On the other hand, (3.4) also implies that for u (at least for a modification of u) it holds that

E sup_{t≤τ∧T} ||u(t, ·) − u_{j}(t, ·)||_{n−2,p}^{p} → 0

for any constant T < ∞. Adding to this that the processes (u_{j}(t, ·), ϕ) are continuous (a.s.) (see Remark 3.2), we conclude that (u(t, ·), ϕ) is also continuous (a.s.). Thus, for any ϕ ∈ C_{0}^{∞}, equality (3.2) not only holds in ^{|}(0, τ]] almost everywhere but also for all t ≤ τ almost surely. Hence, u ∈ H_{p}^{n}(τ) and u_{j} → u in H_{p}^{n}(τ). The theorem is proved.

Remark 3.8. We could replace the first term on the right in (3.3) with ||u||_{ℍ_{p}^{n}(τ)} and, for bounded τ, we would get an equivalent norm by virtue of (3.4). The form of (3.3) that we have chosen is convenient in the future when we need certain constants to be independent of T, see, for instance, Theorem 4.10.

Remark 3.9. In Theorems 7.1 and 7.2 below, we prove much sharper estimates than (3.5).

We also need the following properties of the spaces H_{p}^{n}(τ) and ℍ_{p}^{n}(τ).

Theorem 3.10. Take g ∈ ℍ_{p}^{n}(l_{2}). Then there exists a sequence g_{j} ∈ ℍ_{p}^{n}(l_{2}), j = 1, 2..., such that ||g − g_{j}||_{ℍ_{p}^{n}(l_{2})} → 0 as j → ∞ and

g_{j}^{k} = \{ \begin{matrix}∑_{i=1}^{j} I_{(τ_{i−1}^{j},τ_{i}^{j}]}(t)g_{j}^{ik}(x) & if & k ≤ j, \\ 0 & if & k > j, \end{matrix}

where τ_{i}^{j} are bounded stopping times, τ_{i−1}^{j} ≤ τ_{i}^{j}, and g_{j}^{ik} ∈ C_{0}^{∞}.

Proof. The argument in Remark 3.4 and the fact that C_{0}^{∞} is dense in any H_{p}^{n} show that we only need to consider n = 0. Further, one can easily understand that the set of g ∈ 𝕃_{p}(l_{2}) for which the statement holds forms a linear closed subspace 𝕃 of 𝕃_{p}(l_{2}). We have to prove that 𝕃 = 𝕃_{p}(l_{2}). If this is not true, then, by Riesz’s theorem, there is a nonzero h ∈ 𝕃_{q}(l_{2}) with q = p/(p − 1) such that

E ∫_{0}^{∞} ∫_{ℝ^{d}}(h, g)_{l_{2}} dxdt = 0

for any g ∈ 𝕃. In particular,

E ∫_{0}^{∞} I_{(0,τ]}(∫_{ℝ^{d}} h^{k}g dx)dt = 0

for any bounded stopping time τ, k ≥ 1, and g ∈ C_{0}^{∞}. Since ∫_{ℝ^{d}} h^{k}g dx is equal to a predictable function (a.e.), it follows that ∫_{ℝ^{d}} h^{k}g dx = 0 on ^{|}(0, ∞]] (a.e.). By taking g from a countable subset G in C_{0}^{∞} that is dense in L_{p}, we get that, on a subset of ^{|}(0, ∞]] of full measure,

∫_{ℝ^{d}} h^{k}g dx = 0 ∀g ∈ G, k ≥ 1.

But then h^{k} = 0 on ^{|}(0, ∞]] × ℝ^{d} (a.e.). This contradicts that h ≠ 0 and proves the theorem.

Theorem 3.11. Let T ∈ (0, ∞). If u_{j} ∈ H_{p}^{n}(T), j = 1, 2, ..., and ||u_{j}||_{H_{p}^{n}(T)} ≤ K, where K is a finite constant, then there exists a subsequence j′ and a function u ∈ H_{p}^{n}(T) such that

(i) u_{j′}, u_{j′}(0, ·), 𝔻u_{j′}, and 𝕊u_{j′} converge weakly to u, u(0, ·), 𝔻u, and 𝕊u in ℍ_{p}^{n}(T), L_{p}(Ω, H_{p}^{n−2/p}), ℍ_{p}^{n−2}(T), and ℍ_{p}^{n−1}(T, l_{2}) respectively;

(ii) ||u||_{H_{p}^{n}(T)} ≤ K;

(iii) for any ϕ ∈ C_{0}^{∞} and any t ∈ [0, T] we have (ϕ, u_{j′}(t, ·)) → (ϕ, u(t, ·)) weakly in L_{p}(Ω).

Proof. From properties of L_{p} spaces, it follows that there exists a subsequence j′ such that u_{j′}, u_{j′}(0, ·), 𝔻u_{j′}, 𝕊u_{j′} converge weakly to some u, u_{0}, f, g in ℍ_{p}^{n}(T), L_{p}(Ω, H_{p}^{n−2/p}), ℍ_{p}^{n−2}(T), and ℍ_{p}^{n−1}(T, l_{2}) respectively. Then, for any ϕ ∈ C_{0}^{∞}, the expressions (u_{j′}(t, ·), ϕ), (u_{j′}(0, ·), ϕ), (𝔻u_{j′}(s, ·), ϕ), and (𝕊^{k}u_{j′}(s, ·), ϕ) in the formula

(u_{j′}(t, ·), ϕ) = (u_{j′}(0, ·), ϕ) + ∫_{0}^{t}(𝔻u_{j′}(s, ·), ϕ) ds + ∑_{k=1}^{∞} ∫_{0}^{t}(𝕊^{k}u_{j′}(s, ·), ϕ) dw_{s}^{k}

converge weakly in corresponding spaces. Since integration and stochastic integration can be considered as continuous linear operators and any continuous linear operator is also weakly continuous, we have that, for any ϕ ∈ C_{0}^{∞},

(u(t, ·), ϕ) = (u_{0}, ϕ) + ∫_{0}^{t}(f(s, ·), ϕ) ds + ∑_{k=1}^{∞} ∫_{0}^{t}(g^{k}(s, ·), ϕ) dw_{s}^{k}  (3.8)

for almost all (ω, t) ∈ Ω × [0, T].

By Banach–Saks theorem, there is a sequence (v_{j′}, 𝔻v_{j′}, 𝕊v_{j′}) of convex combinations of (u_{j′}, 𝔻u_{j′}, 𝕊u_{j′}) which converges strongly to (u, f, g) in ℍ_{p}^{n}(T) × ℍ_{p}^{n−2}(T) × ℍ_{p}^{n−1}(T, l_{2}). From (3.4), it follows that

E sup_{t≤T} ||v_{j} − v_{i}||_{n−2,p}^{p} → 0

as i, j → ∞. Therefore, there is a H_{p}^{n−2}–valued function v on Ω × [0, T] such that

E sup_{t≤T} ||v_{j} − v||_{n−2,p}^{p} → 0.

In particular, (v_{j}(t, ·), ϕ) → (v(t, ·), ϕ) uniformly on [0, T] in probability for any ϕ ∈ C_{0}^{∞}. On the other hand, the strong convergence of v_{j} to u in ℍ_{p}^{n}(T) implies that (v_{j}(t, ·), ϕ) → (u(t, ·), ϕ) on Ω × [0, T] in measure. This shows that (v(t, ·), ϕ) = (u(t, ·), ϕ) on Ω × [0, T] (a.e.). Because of arbitrariness of ϕ and the fact that C_{0}^{∞} is dense in the separable spaces conjugate to H_{p}^{m}, u = v (as generalized functions) on Ω × [0, T] (a.e.).

Thus, v ∈ ℍ_{p}^{n}(T). Also, (v_{j}(t), ϕ) are given by equations similar to (3.8), which implies that (v_{j}(t), ϕ) are continuous in t (a.s.). The uniform convergence of (v_{j}(t), ϕ) to (v(t), ϕ) in probability yields the continuity of (v(t), ϕ) (a.s.). By the above, (3.8) holds for almost all (ω, t) ∈ Ω × [0, T] if we replace (u(t, ·), ϕ) by (v(t, ·), ϕ). Since the latter is continuous and the right–hand side of (3.8) is continuous, (v(t, ·), ϕ) equals the right–hand side of (3.8) for all t ∈ [0, T] (a.s.). Hence, v ∈ H_{p}^{n}(T) and we have proved assertion (i) for v instead u, which is irrelevant.

Assertion (ii) follows from the equality u = v on Ω × [0, T] (a.e.) and from the fact that the norm of a weak limit is less than the liminf of norms.

To prove (iii), take ϕ ∈ C_{0}^{∞} and ξ ∈ L_{q}(Ω) with q = p/(p − 1) and write

Eξ(u_{j}(t, ·), ϕ) = Eξ(u_{j}(0, ·), ϕ) + Eξ ∫_{0}^{t}(𝔻u_{j}(s, ·), ϕ) ds + Eξ ∫_{0}^{t}(𝕊^{k}u_{j}(s, ·), ϕ) dw_{s}^{k}.

By what has been said about the properties of the operators of integration and by (i),

lim_{j′→∞} Eξ(u_{j′}(t, ·), ϕ) = lim_{j′→∞}[Eξ(u_{j′}(0, ·), ϕ) + Eξ ∫_{0}^{t}(𝔻u_{j′}(s, ·), ϕ) ds

+ Eξ ∫_{0}^{t}(𝕊^{k}u_{j′}(s, ·), ϕ) dw_{s}^{k}] = Eξ(u(0, ·), ϕ)

+ Eξ ∫_{0}^{t}(𝔻u(s, ·), ϕ) ds + Eξ ∫_{0}^{t}(𝕊^{k}u(s, ·), ϕ) dw_{s}^{k} = Eξ(u(t, ·), ϕ),

which proves (iii) and the theorem.

4. Model Equations

As explained in Sec. 2, to implement our general scheme we have to show that (at least) simple equations are solvable in H_{p,0}^{n}(τ).

Except for Subsec. 4.2, we will always understand equations like (2.1) in the sense of Definition 3.5, which means that we will be looking for a function u ∈ H_{p,0}^{n}(τ) such that

𝔻u = Lu + f, 𝕊u = Λu + g.

In this section, we consider equation (2.1) when b^{i} = c = ν^{k} = 0 and the coefficients a and σ do not depend on x. Throughout the section, we fix real-valued functions a^{ij}(t) and l_{2}-valued functions σ^{i}(t) = {σ^{ik}(t), k ≥ 1} defined for i, j = 1, ..., d on Ω × (0, ∞). Define

α^{ij}(t) = \frac{1}{2}(σ^{i}(t), σ^{j}(t))_{l_{2}}

and assume that a and σ are P-measurable functions, and in the matrix sense

(a^{ij}) = (a^{ij})^{∗}, K(δ^{ij}) ≥ (a^{ij}) ≥ (a^{ij} − α^{ij}) ≥ δ(δ^{ij}),

where K and δ are some fixed strictly positive constants. By the way, the assumption that a > α is necessary even to have L_{2}–theory for SPDEs with constant coefficients (see [30] or Remark 4.8).

Equation (2.1) takes the following form:

du(t, x) = (a^{ij}(t)u_{x^{i}x^{j}}(t, x) + f(t, x)) dt + (σ^{ik}(t)u_{x^{i}}(t, x) + g^{k}(t, x)) dw_{t}^{k}, t > 0. (4.1)

Our plan is as follows. In Subsec. 4.1 we consider the case of the heat equation with random right–hand side and get basic a priori estimates. The results of Subsec. 4.2 show how to reduce general case of equation (4.1) to the case of the heat equation and get some representation formulas for solutions. At that point it turns out that the assumption that u ∈ H_{p}^{n} is not very convenient, and we consider larger spaces. In the final Subsec. 4.3, we present the basic a priori estimates and the existence and uniqueness theorem for equation (4.1).

4.1. Particular Case a^{ij} = δ^{ij}, σ = 0. We start with the equation

du(t, x) = (Δu(t, x) + f(t, x)) dt + g^{k}(t, x) dw_{t}^{k}, t > 0.  (4.2)

We need a lemma from [17] or [20]. Remember that the operators T_{t} are defined by (2.3) and, as always, p ≥ 2.

Lemma 4.1. Let −∞ ≤ a < b ≤ ∞, g ∈ L_{p}((a, b) × ℝ^{d}, l_{2}). Then

∫_{ℝ^{d}} ∫_{a}^{b}[∫_{a}^{t} |∇T_{t−s}g(s, ·)(x)|_{l_{2}}^{2} ds]^{p/2} dt dx ≤ N(d, p) ∫_{ℝ^{d}} ∫_{a}^{b} |g(t, x)|_{l_{2}}^{p} dt dx.

Theorem 4.2. Take f ∈ ℍ_{p}^{−1}, g ∈ 𝕃_{p}(l_{2}). Then

(i) equation (4.2) with zero initial condition has a unique solution u ∈ H_{p}^{1}; (ii) for this solution, we have

||u_{xx}||_{ℍ_{p}^{−1}} ≤ N(d, p)(||f||_{ℍ_{p}^{−1}} + ||g||_{𝕃_{p}(l_{2})});  (4.3)

(iii) for this solution, we have u ∈ C_{loc}([0, ∞), L_{p}) almost surely, and, for any λ, T > 0,

E sup_{t≤T}(e^{−pλt}||u(t, ·)||_{p}^{p}) + E ∫_{0}^{T} e^{−pλt}|| |u|^{(p−2)/p}|u_{x}|^{2/p}(t, ·)||_{p}^{p} dt ≤

N(d, p, λ)(||e^{−λt}f||_{ℍ_{p}^{−1}(T)}^{p} + ||e^{−λt}g||_{𝕃_{p}(T,l_{2})}^{p}).  (4.4)

Proof. It is well known that there exists a continuous linear operator

P : H_{p}^{−1} → (L_{p})^{d+1}

such that if h ∈ H_{p}^{−1} and P h = (h_{0}, h̃^{1}, ..., h̃^{d}), then h = h_{0} + div h̃ and

||h̃||_{p} + ||h_{0}||_{p} ≤ N(d, p)||h||_{−1,p}, ||h||_{−1,p} ≤ N(d, p){||h̃||_{p} + ||h_{0}||_{p}}.  (4.5)

Actually, one can take h̃ = −grad (1 − Δ)^{−1}h and h_{0} = h − div h̃ = (1 − Δ)^{−1}h. Indeed, ||h_{0}||_{p} = ||h||_{−2,p} ≤ ||h||_{−1,p}. Also, the fact that ∂/∂x^{i} is a bounded operator from H_{p}^{n} to H_{p}^{n+1} for any n (see, for instance, [33]) means that (∂/∂x^{i})(1 − Δ)^{−1/2} is a bounded operator from H_{p}^{n} to H_{p}^{n} and (∂/∂x^{i})(1 − Δ)^{−1} is a bounded operator from H_{p}^{n} to H_{p}^{n−1}. This is why ||h̃||_{p} ≤ N(d, p)||h||_{−1,p}. This gives the first estimate in (4.5). On the other hand, (1 − Δ)^{−1/2}h = (1 − Δ)^{−1/2}h_{0} + div (1 − Δ)^{−1/2}h̃, and both operators (1 − Δ)^{−1/2} and div (1 − Δ)^{−1/2} are bounded on L_{p}.

Define (f_{0}, f̃) = P f. Then equation (4.2) takes the form

du = (Δu + f_{0} + div f̃) dt + g^{k} dw_{t}^{k},  (4.6)

and we supply it with zero initial condition. We will prove that, for arbitrary f_{0}, f̃^{i} ∈ 𝕃_{p}, our assertions hold for (4.6) in place of (4.2). Of course, in (4.3) and (4.4), by f we mean f_{0} + div f̃.

A particular case. First we consider the case in which

\begin{matrix}f_{0}(t, x) = ∑_{i=1}^{m} I_{(τ_{i−1},τ_{i}]}(t)f_{0i}(x), f̃(t, x) = ∑_{i=1}^{m} I_{(τ_{i−1},τ_{i}]}(t)f̃_{i}(x), \\ g(t, x) = ∑_{k=1}^{m} g^{k}(t, x)h_{k}, g^{k}(t, x) = ∑_{i=1}^{m} I_{(τ_{i−1},τ_{i}]}(t)g^{ik}(x), \end{matrix}  (4.7)

where (h_{k}) is the standard orthonormal basis in l_{2}, m < ∞, τ_{i} are bounded stopping times, τ_{i−1} ≤ τ_{i}, and f_{0i}, f̃_{i}, g^{ik} ∈ C_{0}^{∞}.

Set

v(t, x) = ∫_{0}^{t} g^{k}(s, x) dw_{s}^{k} = ∑_{i,k=1}^{m} g^{ik}(x)(w_{t∧τ_{i}}^{k} − w_{t∧τ_{i−1}}^{k}),

u(t, x) = v(t, x) + ∫_{0}^{t} T_{t−s}[Δv + f](s, ·)(x) ds, ∀t ≥ 0.  (4.8)

As easy to see, the function u − v is infinitely differentiable in (t, x) and satisfies the equation

\frac{∂z}{∂t} = Δz + Δv + f.

It follows that, for any x, the function u(t, x) satisfies almost surely the following form of (4.6):

u(t, x) = ∫_{0}^{t}(Δu(s, x) + f(s, x)) ds + ∑_{k=1}^{m} ∫_{0}^{t} g^{k}(s, x) dw_{s}^{k}.  (4.9)

Next, we want to obtain some bounds on norms of u. Let

u_{1}(t, x) = ∫_{0}^{t} T_{t−s}f(s, x) ds.

According to Theorem 2.1 (for any ω),

||u_{1xx}||_{L_{p}(ℝ_{+},H_{p}^{−1})} ≤ N||f||_{L_{p}(ℝ_{+},H_{p}^{−1})}.  (4.10)

Furthermore, use again that the operators (∂/∂x^{i})(1 − Δ)^{−1/2} are bounded in L_{p} for any p > 1. Then

||u_{xx} − u_{1xx}||_{ℍ_{p}^{−1}}^{p} ≤ N||u_{x} − u_{1x}||_{𝕃_{p}}^{p} = N ∫_{0}^{∞} ∫_{ℝ^{d}} E|u_{x} − u_{1x}|^{p}(t, x) dx dt.  (4.11)

To make some further transformations of this formula, we note that if z^{k} = z^{k}(x) are bounded Borel functions, then, by Itô’s formula applied to the increment over [0, t] of

(∫_{r}^{t} T_{t−s}z^{k} ds)(w_{r∧τ_{2}}^{k} − w_{r∧τ_{1}}^{k})

(with no summation in k) as a function of r, we obtain (a.s.)

0 = − ∫_{0}^{t}(w_{r∧τ_{2}}^{k} − w_{r∧τ_{1}}^{k})T_{t−r}z^{k} dr + ∫_{0}^{t} I_{(τ_{1},τ_{2}]}(r)(∫_{r}^{t} T_{t−s}z^{k} ds) dw_{r}^{k}.

By using this for our particular g, or by using the stochastic version of the Fubini theorem and coming back to (4.8), for any t ≥ 0 and x ∈ ℝ^{d}, we get

u_{x}(t, x) − u_{1x}(t, x) = v_{x}(t, x) + ∫_{0}^{t} T_{t−s} ∑_{k=1}^{m} ∫_{0}^{s} Δg_{x}^{k}(r, x) dw_{r}^{k} ds

= v_{x}(t, x) − ∑_{k=1}^{m} ∫_{0}^{t} ∫_{r}^{t} \frac{d}{ds}T_{t−s}g_{x}^{k}(r, x) ds dw_{r}^{k} = ∑_{k=1}^{m} ∫_{0}^{t} T_{t−r}g_{x}^{k}(r, x) dw_{r}^{k} (a.s.).

Hence, by the Burkholder–Davis–Gundy inequality,

E|u_{x} − u_{1x}|^{p}(t, x) ≤ NE[∫_{0}^{t} ∑_{k=1}^{m} |T_{t−r}g_{x}^{k}(r, x)|^{2} dr]^{p/2}

= NE[∫_{0}^{t} |T_{t−r}g_{x}(r, x)|_{l_{2}}^{2} dr]^{p/2}.

By plugging this into (4.11) and by applying Lemma 4.1, we obtain

||u_{x} − u_{1x}||_{𝕃_{p}}^{p} ≤ NE ∫_{0}^{∞} ∫_{ℝ^{d}}[∫_{0}^{t} |∇T_{t−s}g(s, x)|_{l_{2}}^{2} ds]^{p/2} dx dt ≤ N||g||_{𝕃_{p}(l_{2})}^{p}.

This along with (4.10) gives us (4.3). However, we do not know yet that u ∈ H_{p}^{1}.

Our next step is to prove (4.4) for sufficiently large λ. We repeat briefly the corresponding arguments from [25] or [30]. From (4.9) and Itô’s formula,

|u(t, x)|^{p}e^{−λt}

= ∫_{0}^{t} e^{−λs}(p|u|^{p−2}uΔu + p|u|^{p−2}uf + \frac{1}{2}p(p − 1)|u|^{p−2}|g|_{l_{2}}^{2} − λ|u|^{p})(s, x) ds

+ p ∑_{k≤m} ∫_{0}^{t} e^{−λs}|u|^{p−2}ug^{k}(s, x) dw_{s}^{k}.

We integrate here with respect to x and use the stochastic Fubini theorem and the fact that u(t, x), g(t, x), and their derivatives decrease very fast when |x| → ∞. Then we integrate by parts in ∫ |u|^{p−2}uΔu dx and also notice that, for q = p/(p − 1),

∫_{ℝ^{d}} |u|^{p−2}uf(s, x) dx = −(p − 1) ∫_{ℝ^{d}} |u|^{p−2}u_{x} · f̃(s, x) dx + ∫_{ℝ^{d}} |u|^{p−2}uf_{0}(s, x) dx,

| ∫_{ℝ^{d}} |u|^{p−2}u_{x} · f̃(s, x) dx| ≤ ∫_{ℝ^{d}}(|u|^{(p−2)/2}|u_{x}|)^{q}|u|^{q(p−2)/2} dx

+ ||f̃(s, ·)||_{p}^{p} ≤ N||f(s, ·)||_{−1,p}^{p} + N_{1}||u(s, ·)||_{p}^{p} + \frac{1}{2} || |u|^{(p−2)/p}|u_{x}|^{2/p}(s, ·)||_{p}^{p},

∫_{ℝ^{d}} |u(s, x)|^{p−2}u(s, x)f_{0}(s, x) dx ≤ ||f_{0}(s, ·)||_{p}^{p} + ||u(s, ·)||_{p}^{p}

≤ N||f(s, ·)||_{−1,p}^{p} + ||u(s, ·)||_{p}^{p}.

In this way, for λ ≥ p(p − 1)N_{1} + p + p(p − 1)/2, we get

||u(t, ·)||_{p}^{p}e^{−λt} + \frac{p(p − 1)}{2} ∫_{0}^{t} || |u|^{(p−2)/p}|u_{x}|^{2/p}(s, ·)||_{p}^{p}e^{−λs} ds

≤ N ∫_{0}^{t}[||f(s, ·)||_{−1,p}^{p} + ||g(s, ·)||_{p}^{p}]e^{−λs} ds

+ p ∑_{k≤m} ∫_{0}^{t} e^{−λs}{∫_{ℝ^{d}} |u|^{p−2}ug^{k}(s, x) dx} dw_{s}^{k},

where N = N(p). After this, basically, one takes expectations and applies certain standard transformations based on the Burkholder–Davis–Gundy inequalities. For more detail we refer the reader to [25] or [30].

Finally, the assertion about the arbitrariness of λ in (4.4) can be easily justified by rescaling arguments when instead of f, g, and w one takes (c^{2}f, cg)(c^{2}t, cx) and c^{−1}w_{c^{2}t} and gets u(c^{2}t, cx) instead of u(t, x).

From our explicit formulas and from the particular choice of f and g, it also follows that u ∈ C_{loc}([0, ∞), H_{p}^{n}) for any n (and for any ω). This proves (iii).

From estimates (4.3) and (4.4), we conclude that u ∈ ⋂_{T >0} ℍ_{p}^{1}(T). Furthermore, from the pointwise equation (4.9) by the stochastic Fubini theorem it follows easily that u solves (4.2) in the sense of Definition 3.1. Hence u ∈ H_{p}^{1}, which proves a part of assertion (i). The uniqueness in (i) follows from the fact that for f ≡ 0, g ≡ 0 we have the heat equation and the uniqueness of its solution in our class of functions is a standard fact (we say more about this in the proof of Lemma 4.9). This completes the proof in the case of step functions f, g.

General case. In the case of general f, g, we observe that the uniqueness in (i) is proved as above. As far as other assertions are concerned we are going to use Theorem 3.10 and Remark 3.6.

If we consider all functions f_{0}, f̃^{i}, g^{k} as one sequence, then, by Theorem 3.10, we can approximate them by functions f_{0j}, f̃_{j}^{i}, g_{j}^{k} of type (4.7). Let u_{j} be the corresponding solutions of (4.6). By the result for the particular case, {u_{j}} is a Cauchy sequence in H_{p}^{1}, and, by Theorem 3.7, there is a u ∈ H_{p}^{1} to which u_{j} converges in H_{p}^{1}. Remark 3.6 and the convergence ||u_{xx} − u_{jxx}||_{ℍ_{p}^{−1}} → 0 prove that 𝔻u = Δu + f and 𝕊u = g. In particular, this proves assertion (i).

Assertion (ii) follows from the construction of u. From assertion (iii) available in the particular case, we get that u_{j} is a Cauchy sequence in L_{p}(Ω, C([0, T], L_{p})) for any T. Therefore, it converges in this space to a function ū. It follows easily that, for any ϕ ∈ C_{0}^{∞},

(ū(t, ·), ϕ) = ∫_{0}^{t}{(ū(s, ·), Δϕ) + (f(s, ·), ϕ)} ds + ∫_{0}^{t}(g^{k}(s, ·), ϕ) dw_{s}^{k}

for all t (a.s.). Therefore, u − ū is a generalized solution of the heat equation with zero initial condition and with bounded L_{p}–norm (a.s.). This implies that ||(u − ū)(t, ·)||_{p} = 0 for all t (a.s.), so that u ∈ C([0, T], L_{p}) for all T (a.s.). Finally, we get (4.4) by Fatou’s lemma taking into account that

∫_{0}^{T} ∫_{ℝ^{d}} |∇(u − u_{j})|^{p} dxdt = ∫_{0}^{T} ∫_{ℝ^{d}} |∇(1 − Δ)^{−1/2}(1 − Δ)^{1/2}(u − u_{j})|^{p} dxdt

≤ N ∫_{0}^{T} ∫_{ℝ^{d}} |(1 − Δ)^{1/2}(u − u_{j})|^{p} dxdt → 0

in probability for any T. The theorem is proved.

Remark 4.3. Although (2.7) holds for all p ∈ (1, ∞), it follows from [17] that Lemma 4.1 is false if p < 2.

4.2. Relation of the Solutions of (4.1) to the Solutions of the Heat Equation. It turns out that the investigation of general equation (4.1) with coefficients independent of x can be quite formally reduced to the particular case of the heat equation. First, we explain how to do this without caring about rigorousness, and then proceed with formal proofs.

The first observation consists of the following. Assume that we have

du(t, x) = f(t, x) dt + g^{k}(t, x) dw_{t}^{k},  (4.12)

and we define a process x_{t} and a function v by

x_{t}^{i} = ∫_{0}^{t} σ^{ik}(s) dw_{s}^{k}, i = 1, ..., d, v(t, x) = u(t, x − x_{t}).  (4.13)

Then, by applying formally the Itô–Wentzell formula, we get

\begin{matrix}dv(t, x) = & [f(t, x − x_{t}) + α^{ij}(t)v_{x^{i}x^{j}}(t, x) − (g_{x^{i}}(t, x − x_{t}), σ^{i}(t))_{l_{2}}] dt \\ + [g^{k}(t, x − x_{t}) − v_{x^{i}}(t, x)σ^{ik}(t)] dw_{t}^{k}. \end{matrix}  (4.14)

This shows how to introduce the terms v_{x^{i}}σ^{ik} in equation (4.12) and also shows again a kind of necessity for g to have the first derivatives in x.

This device alone is not sufficient, since, if we had Δu + f̄ instead of f in (4.12), then, in (4.14), we would get the second order differential operator (δ^{ij} + α^{ij})∂^{2}/∂x^{i}∂x^{j} with coefficients strongly related to the coefficients of v_{x^{i}}σ^{ik}(t). We could get around this difficulty if we manage to start with equations with more general operators L instead Δ. Here the second observation comes that if, instead of (4.12), we consider

du(t, x) = (Δu + f̄) dt + g^{k}(t, x) dw_{t}^{k},

and take expectations in the counterpart of (4.14) corresponding to this equation, then, assuming that σ is nonrandom, we get indeed an equation for Ev(t, x) with operator L different from Δ. By the way, this method of studying parabolic equations with coefficients independent of x was applied in [18] in order to show that “whatever” estimate is true for the heat equation, it is also true for any parabolic equation with coefficients independent of x. Of course, taking expectations “kills” all randomness in the equation, and therefore we use a conditional expectation.

Definition 4.4. Denote by 𝔇 the set of all D-valued functions u (written as u(t, x) in a common abuse of notation) on Ω × [0, ∞) such that, for any ϕ ∈ C_{0}^{∞},

(i) the function (u, ϕ) is P–measurable, (ii) for any ω ∈ Ω and T ∈ (0, ∞), we have

∫_{0}^{T} sup_{x∈ℝ^{d}} |(u(t, ·), ϕ(· − x))|^{2} dt < ∞.  (4.15)

In the same way, we define 𝔇(l_{2}) by considering l_{2}-valued linear functionals on C_{0}^{∞} and replacing | · | in (4.15) by | · |_{l_{2}}.

Remark 4.5. Notice that (u(t, ·), ϕ(· − x)) is continuous in x and Borel in t so that (4.15) makes sense. Also, for p ≥ 2, q = p/(p − 1), and any n,

∫_{0}^{T} sup_{x∈ℝ^{d}} |(u(t, ·), ϕ(· − x))|^{2} dt ≤ ∫_{0}^{T} ||(u(t, ·)||_{n,p}^{2}||ϕ||_{−n,q}^{2} dt

≤ ||ϕ||_{−n,q}^{2}T^{(p−2)/p} (∫_{0}^{T} ||(u(t, ·)||_{n,p}^{p} dt)^{2/p} . (4.16)

This shows that if u ∈ H_{p}^{n}, then condition (4.15) is satisfied at least for almost all ω. Also, if u ∈ H_{p}^{n}, then (3.2) holds, which shows that (u(t, ·), ϕ) is indistinguishable from a predictable process. This is true for any ϕ ∈ C_{0}^{∞}. From separability of H_{q}^{−n}, it follows that we can modify u on a set of probability zero and after this we get a function belonging to 𝔇. This is the sense in which we write

H_{p}^{n} ⊂ 𝔇.  (4.17)

Definition 4.6. Let f, u ∈ 𝔇, g ∈ 𝔇(l_{2}). We say that the equality

du(t, x) = f(t, x) dt + g(t, x) dw_{t}, t > 0,  (4.18)

holds in the sense of distributions if for any ϕ ∈ C_{0}^{∞}, with probability 1 for all t ≥ 0 we have

(u(t, ·), ϕ) = (u(0, ·), ϕ) + ∫_{0}^{t}(f(s, ·), ϕ) ds + ∫_{0}^{t}(g^{k}(s, ·), ϕ) dw_{s}^{k}.  (4.19)

Observe that, since |(g, ϕ)(t)|_{l_{2}}^{2} is locally summable in t, the last series in (4.19) converges uniformly in t on every finite interval of time in probability.

In this subsection, we always understand equation (4.1) in the sense of distributions. Notice that if u ∈ H_{p}^{n} and u satisfies (4.19), then, by (4.17), u ∈ 𝔇 and (4.18) holds in the sense of distributions. An advantage of Definition 4.6 is that one need not check summability of any derivative.

Lemma 4.7. Let f, u ∈ 𝔇, g ∈ 𝔇(l_{2}). Assume the definitions in (4.13). Then (4.12) holds (in the sense of distributions) if and only if (4.14) holds (in the sense of distributions).

Proof. First remember that, for a distribution α(x) and y ∈ ℝ^{d}, by α(x − y) we mean the distribution defined by (α, ϕ(· + y)). Also from relations like (cf. (4.16))

∫_{0}^{T} sup_{y∈ℝ^{d}} |(v_{xx}(t, ·), ϕ(· − y))|^{2} dt = ∫_{0}^{T} sup_{y∈ℝ^{d}} |(v(t, ·), ϕ_{xx}(· − y))|^{2} dt

= ∫_{0}^{T} sup_{y∈ℝ^{d}} |(u(t, ·), ϕ_{xx}(· + x_{t} − y))|^{2} dt = ∫_{0}^{T} sup_{y∈ℝ^{d}} |(u(t, ·), ϕ_{xx}(· − y))|^{2} dt < ∞,

∫_{0}^{T} sup_{y∈ℝ^{d}} |((g_{x^{i}}(t, · − x_{t}), σ^{i}(t))_{l_{2}}, ϕ(· − y))|^{2} dt

= ∫_{0}^{T} sup_{y∈ℝ^{d}} |((g_{x^{i}}(t, · − x_{t}), ϕ(· − y)), σ^{i}(t))_{l_{2}}|^{2} dt

≤ ∫_{0}^{T} |σ^{i}(t)|_{l_{2}}^{2} dt ∫_{0}^{T} sup_{y∈ℝ^{d}} |(g_{x^{i}}(t, · − x_{t}), ϕ(· − y))|_{l_{2}}^{2} dt < ∞

it follows that v(t, x), f(t, x − x_{t}) and (g_{x^{i}}(t, x − x_{t}), σ^{i}(t))_{l_{2}} belong to 𝔇 and g(t, x − x_{t}) and v_{x^{i}}(t, x)σ^{i}(t) belong to 𝔇(l_{2}). Furthermore, for any ϕ ∈ C_{0}^{∞}, the function F (t, x) := (u(t, · − x), ϕ) has a stochastic differential in t for any x and is infinitely differentiable with respect to x. Now our assertion immediately follows from the Itô–Wentzell formula for F (t, x_{t}). By the way, one can easily memorize (and perhaps prove) this formula by considering the following computations:

d ∫_{ℝ^{d}} u(t, x − x_{t})ϕ(x) dx = d ∫_{ℝ^{d}} u(t, x)ϕ(x + x_{t}) dx = ∫_{ℝ^{d}} ϕ(x + x_{t})du(t, x) dx
+ ∫_{ℝ^{d}} u(t, x)dϕ(x + x_{t}) dx + ∫_{ℝ^{d}}(du(t, x))dϕ(x + x_{t}) dx
= {∫_{ℝ^{d}} ϕ(x + x_{t})f(t, x) dx + ∫_{ℝ^{d}} u(t, x)α^{ij}(t)ϕ_{x^{i}x^{j}}(x + x_{t}) dx
+ ∫_{ℝ^{d}} g^{k}(t, x)ϕ_{x^{i}}(x + x_{t})σ^{ik}(t) dx} dt
+ {∫_{ℝ^{d}} ϕ(x + x_{t})g^{k}(t, x) dx + ∫_{ℝ^{d}} u(t, x)ϕ_{x^{i}}(x + x_{t})σ^{ik}(t) dx} dw_{t}^{k}
= ∫_{ℝ^{d}} {f(t, x − x_{t}) + α^{ij}(t)u_{x^{i}x^{j}}(t, x − x_{t}) − g_{x^{i}}^{k}(t, x − x_{t})σ^{ik}(t)} ϕ(x) dx dt
+ ∫_{ℝ^{d}} {g^{k}(t, x − x_{t}) − u_{x^{i}}(t, x − x_{t})σ^{ik}(t)} ϕ(x) dx dw_{t}^{k}.

The lemma is proved.

Remark 4.8. If, instead of (4.12), u satisfies the equation

du(t, x) = (a^{ij}(t)u_{x^{i}x^{j}}(t, x) + h(t, x)) dt + σ^{ik}(t)u_{x^{i}}(t, x) dw_{t}^{k},

then (4.14) takes the form

\frac{∂}{∂t}v(t, x) = ((a^{ij}(t) − α^{ij}(t))v_{x^{i}x^{j}}(t, x) + h(t, x − x_{t}) t > 0,  (4.20)

and can be considered on each ω separately. Observe that if a(t) < α(t), then the initial– value problem v(0) = v_{0} for equation (4.20) is ill posed.

This shows that operators appearing in the stochastic term should be subordinated in a certain sense to the operator in the deterministic part of equation. This is needed in order to construct an L_{p}–theory. On the other hand, take d = 1 and a one-dimensional Wiener process w_{t}. Consider the following equation

du(t, x) = iu′_{x}(t, x) dw_{t}.

Surprisingly enough and somewhat in spite of what is said above, this equation has a very nice solution for each initial data u_{0} ∈ L_{2}. One gets the solution after passing to Fourier transforms. It turns out that ũ(t, ξ) = u_{0}(ξ) exp{ξw_{t} − (1/2)|ξ|^{2}t}. The function ũ(t, ξ) decays vary fast when |ξ| → ∞, which shows that u(t, x) is infinitely differentiable in x. Also notice that, taking expectations, we see that Eu(t, x) = u_{0}(x) if u_{0} is nonrandom, and in this case we get a representation of any L_{2} function as an integral over Ω of functions u(ω, 1, x) which are infinitely differentiable in x. However, the major drawback of such equations is that E|u(t, 0)|^{p} = ∞ for any p > 1 if, for example, ũ_{0}(ξ) ≥ exp{−λ|ξ|}, where λ is a constant.

Lemma 4.9. Let f ∈ 𝔇, g ∈ 𝔇(l_{2}), u_{0} be a D-valued function on Ω. Then the following assertions hold:

(i) In 𝔇 there can exist only one (up to evanescence) solution of equation (4.1) with the initial condition u(0, ·) = u_{0}.

(ii) Let F_{t} = W_{t} ∨ B_{t} for t ≥ 0, and assume that σ–fields W_{t} and B_{t} form independent increasing filtrations. Let W and B be sets such that W ∪ B = {1, 2...}. Assume that (w_{t}^{k}, W_{t}) and (w_{t}^{r}, B_{t}) are Wiener processes for k ∈ W, r ∈ B. Let u ∈ 𝔇 satisfy equation (4.1) (in the sense of distributions), and let a, f, σ, g be W_{t}–adapted. Finally, assume that there exists an n ∈ (−∞, ∞) such that f ∈ ℍ_{2}^{n}(T), g ∈ ℍ_{2}^{n}(T, l_{2}) for any T ∈ (0, ∞) and u(0, ·) is W_{0}–measurable and

E||u(0, ·)||_{n,2}^{2} < ∞.

Then in 𝔇 there exists a unique solution ũ of the equation

dũ = (a^{ij}ũ_{x^{i}x^{j}} + f) dt + ∑_{k∈W}(σ^{ik}ũ_{x^{i}} + g^{k}) dw_{t}^{k}, t > 0.  (4.21)

In addition, for any ϕ ∈ C_{0}^{∞} and t ≥ 0,

(ũ(t, ·), ϕ) = E{(u(t, ·), ϕ)|W_{t}}  (a.s.).  (4.22)

Proof. (i) As always, we can take f ≡ 0, g ≡ 0, and u_{0} ≡ 0, and, by Lemma 4.7, it suffices to consider only the case σ ≡ 0. In this case, for any given ϕ ∈ C_{0}^{∞} we have

(u(t, ·), ϕ) = ∫_{0}^{t}(u(s, ·), L(s)ϕ) ds, t ≥ 0,

almost surely. Putting here ϕ(·−x) instead of ϕ and observing that both sides are continuous and bounded in (t, x) on [0, T] × ℝ^{d} for any T < ∞ (cf. (4.15)), we get that the function F (t, x) := (u(t, ·), ϕ(· − x)) is bounded in (t, x) on [0, T] × ℝ^{d} for any T < ∞, infinitely differentiable in x, and satisfies the equation

F (t, x) = ∫_{0}^{t} L(s)F (s, x) ds ∀t, x

(a.s.). From the theory of parabolic equations, it follows that F (t, x) = 0 for all t, x (a.s.). This means that (u(t, ·), ϕ) = 0 for all t (a.s.). Now take ϕ with unit integral. Then for any x and n with probability 1, (u(t, ·), n^{d}ϕ(n(· − x))) = 0 for all t. Since this function is continuous in x, we have (u(t, ·), n^{d}ϕ(n(·−x))) = 0 for all t and x with probability 1. Finally, (u(t, ·), n^{d}ϕ(n(·−x))) → u(t, x) as n → ∞ for all (ω, t, x) in the sense of distributions, which implies that, with probability 1, we have u(t, ·) = 0 for all t as stated.

(ii) First, notice that, according to [30], equation (4.1) has a unique solution v in the space ℍ_{2}^{n+1}(T) for any T. The definition of solutions in ℍ_{2}^{n+1}(T) from [30] is slightly different, but v is continuous (a.s.) as an H_{2}^{n}-valued process and

E sup_{t≤T} ||v(t, ·)||_{n,2}^{2} < ∞  ∀T < ∞,  (4.23)

so that v is a 𝔇-solution of (4.1). It follows from (i) that our function u coincides with v and therefore belongs to ℍ_{2}^{n+1}(T) for any T, and (4.23) holds for u. Furthermore, with probability 1 for all t at once,

u(t) = u(0) + ∫_{0}^{t}[a^{ij}(s)u_{x^{i}x^{j}}(s) + f(s)] ds + ∫_{0}^{t}[σ^{ik}(s)u_{x^{i}}(s) + g(s)] dw_{s}^{k},

where all integrals are taken in the sense of the Hilbert space H_{2}^{n−1} (see [30]). By Theorem 1.4.7 of [30], or rather by its Hilbert-space counterpart, there exists an H_{2}^{n+1}-valued , W_{t}- predictable function ū(t) such that, for almost any t, we have

ū(t) = E{u(t)|W_{t}}, ū_{x}(t) = E{u_{x}(t)|W_{t}}, ū_{xx}(t) = E{u_{xx}(t)|W_{t}} (a.s.)

(conditional expectations of Hilbert–space valued random elements) and

ū(t) = u(0) + ∫_{0}^{t}[a^{ij}(s)ū_{x^{i}x^{j}}(s) + f(s)] ds + ∑_{k∈W} ∫_{0}^{t}[σ^{ik}(s)ū_{x^{i}}(s) + g^{k}(s)] dw_{s}^{k} (4.24)

for almost all t and ω. The right-hand side here is a continuous H_{2}^{n−1}-valued process which we denote by ũ and we show that ũ is the function we need.

By definition and by the equality ū = ũ (a.e.), ũ satisfies (4.24) for all t with probability 1 and also is a continuous process in H_{2}^{n−1}. This implies that ũ ∈ 𝔇 and ũ is a solution of (4.21). To prove (4.22) for any t, it remains only to observe that, again by Theorem 1.4.7 of [30], the conditional expectation E{u(t)|W_{t}} for any t is equal to the right-hand side of (4.24) almost surely. The lemma is proved.

4.3. General Equation (4.1) with Coefficients Independent of x.

Theorem 4.10. Take n ∈ ℝ and let f ∈ ℍ_{p}^{n−1}, g ∈ ℍ_{p}^{n}(l_{2}). Then

(i) equation (4.1) with zero initial condition has a unique solution u ∈ H_{p}^{n+1}; (ii) for this solution, we have

||u_{xx}||_{ℍ_{p}^{n−1}} ≤ N(||f||_{ℍ_{p}^{n−1}} + ||g||_{ℍ_{p}^{n}(l_{2})}), ||u||_{H_{p}^{n+1}} ≤ N||(f, g)||_{F_{p}^{n−1}},  (4.25)

where N = N(d, p, δ, K); (iii) we have u ∈ C_{loc}([0, ∞), H_{p}^{n}) almost surely and for any λ, T > 0,

E sup_{t≤T}(e^{−pλt}||u(t, ·)||_{n,p}^{p}) ≤ N(||e^{−λt}f||_{ℍ_{p}^{n−1}(T)}^{p} + ||e^{−λt}g||_{ℍ_{p}^{n}(T,l_{2})}^{p}),  (4.26)

where N = N(d, p, δ, K, λ).

Proof. Since one can apply the operator (I − Δ)^{n/2} to both sides of (4.1), it suffices to prove the theorem only for n = 0. Furthermore, as we have already noticed in (4.17), any function u ∈ H_{p}^{1} also belongs to 𝔇. This and Lemma 4.9 prove the uniqueness in (i). Also, the fact that our norms are translation invariant, combined with Lemma 4.7, shows that, to prove the existence in (i) and all other assertions of the theorem, we only need to consider the case σ ≡ 0. As in the proof of Theorem 4.2, we can assume that f and g are as in (4.7). In this case, as we know from [25] or [30], equation (4.1) has a unique 𝔇–solution u that belongs to C_{b}([0, T] × ℝ^{d}) and C([0, T], L_{2}) almost surely for any T < ∞. It follows that u ∈ C([0, T], L_{p}) almost surely for any T < ∞. Estimate (4.26) also follows from [25] or [30] as in the proof of Theorem 4.2. It remains only to prove that u ∈ H_{p}^{1} and that (4.25) holds. Since u is a 𝔇–solution, to prove that u ∈ H_{p}^{1}, it suffices to prove that u ∈ ℍ_{p}^{1}(T) for any T < ∞.

Since the matrix a is uniformly non-degenerate, by making a nonrandom time change, we can reduce the general case to the case a ≥ I. In this case, define the matrix-valued function σ̄(t) = σ̄^{∗}(t) ≥ 0 as a solution of the equation σ̄^{2}(t) + 2I = 2a(t). Furthermore, without loss of generality, we assume that on our probability space we are also given a d-dimensional Wiener process B_{t} independent of F_{t}.

Now, consider the equation

dv(t, x) = [Δv(t, x) + f(t, x − ∫_{0}^{t} σ̄(s) dB_{s})] dt + g^{k}(t, x − ∫_{0}^{t} σ̄(s) dB_{s}) dw_{t}^{k}  (4.27)

with zero initial condition. Replace the predictable σ–field P with predictable σ–field generated by F_{t} ∨ σ(B_{s}; s ≤ t). Then the spaces H_{p}^{n} become larger. By Theorem 4.2 there is a solution v of (4.27) possessing properties (i) through (iii) listed in Theorem 4.2 (with new H_{p}^{1}). Use again that, after changing, if necessary, v on a set of probability zero, the function v becomes a 𝔇–solution of (4.27). Then, by Lemma 4.7, the function z(t, x) := v(t, x + ∫_{0}^{t} σ̄(s) dB_{s}) is a 𝔇–solution of

dz(t, x) = (a^{ij}(t)z_{x^{i}x^{j}}(t, x) + f(t, x)) dt + g^{k}(t, x) dw_{t}^{k} + z_{x^{i}}(t, x)σ̄^{ij}(t) dB_{t}^{j},

and by Lemma 4.9 there is a solution ũ ∈ 𝔇 of

dũ(t, x) = (a^{ij}(t)ũ_{x^{i}x^{j}}(t, x) + f(t, x)) dt + g^{k}(t, x) dw_{t}^{k},

which is (4.1) in our case. In addition, for any ϕ ∈ C_{0}^{∞} and t ≥ 0,

(ũ(t, ·), ϕ) = E{(z(t, ·), ϕ)|F_{t}} = E{(v(t, · + ∫_{0}^{t} σ̄(s) dB_{s}), ϕ)|F_{t}} (a.s.).

In particular, it follows from this equality that ũ is a 𝔇–solution with respect to the initial predictable σ–field P, and from uniqueness we get ũ = u. Therefore,

(u(t, ·), ϕ) = E{(v(t, · + ∫_{0}^{t} σ̄(s) dB_{s}), ϕ)|F_{t}} (a.s.).  (4.28)

It follows that

|(u(t, ·), ϕ)|^{p} ≤ E{||v(t, ·)||_{1,p}^{p}|F_{t}}||ϕ||_{−1,q}^{p}  (4.29)

(a.s.) for any ϕ ∈ C_{0}^{∞} and t ≥ 0, where q = p/(p − 1).

Next, take a countable family Φ ⊂ C_{0}^{∞} dense in C_{0}^{∞}. Observe that, given a distribution ψ, we have ψ ∈ H_{p}^{1} if and only if for any ϕ ∈ Φ we have |(ψ, ϕ)| ≤ N||ϕ||_{−1,q} with a constant N independent of ϕ. Indeed, in this case there exists a bounded linear functional ℓ on H_{q}^{−1} such that ℓ(ϕ) = (ψ, ϕ) for any ϕ ∈ Φ. Since ℓ(ϕ) = ((1 − Δ)^{−1/2}h, ϕ) with an h ∈ L_{p} and Φ is dense in C_{0}^{∞}, we have ψ = (1 − Δ)^{−1/2}h ∈ H_{p}^{1} indeed. In particular, this fact implies that the set {(ω, t) : w(ω, t, ·) ∈ H_{p}^{1}} is measurable (even predictable) for any w ∈ 𝔇, say for w = u.

We also know that v ∈ H_{p}^{1}, which implies that E||v(t, ·)||_{1,p}^{p} < ∞ for almost all t. Fix such a t. Then there exists a set Ω′ of probability 1 such that E{||v(t, ·)||_{1,p}^{p}|F_{t}} < ∞ on Ω′ and (4.29) holds for all ω ∈ Ω′ and ϕ ∈ Φ. Hence, u(t, ·) ∈ H_{p}^{1} for the chosen t and all ω ∈ Ω′. In particular, u(t, ·) ∈ H_{p}^{1} for almost all (ω, t), and from (4.29) it follows that

||u(t, ·)||_{1,p}^{p} ≤ E{||v(t, ·)||_{1,p}^{p}|F_{t}} (a.s.), ||u||_{ℍ_{p}^{1}(T)} ≤ ||v||_{ℍ_{p}^{1}(T)} < ∞.

Thus u ∈ ℍ_{p}^{1}(T) for any T < ∞ indeed and u ∈ H_{p}^{1}.

Similarly, from the equality

(u_{xx}(t, ·), ϕ) = E{(v_{xx}(t, · + ∫_{0}^{t} σ̄(s) dB_{s}), ϕ)|F_{t}} (a.s.),

one gets

||u_{xx}(t, ·)||_{−1,p}^{p} ≤ E{||v_{xx}(t, ·)||_{−1,p}^{p}|F_{t}} (a.s.).

This and the properties of v immediately yield (4.25). The theorem is proved.

Remark 4.11. By using the self-similarity of equation (4.1), it is possible to obtain further estimates from estimates like (4.26) . For instance, remembering that H_{p}^{1} = W_{p}^{1}, one sees that, for n = 1 and λ = 1/p, estimate (4.26) implies that

E sup_{t≤T}{||u_{x}(t, ·)||_{p}^{p} + ||u(t, ·)||_{p}^{p}} ≤ N(d, p, δ, K)e^{T} (||f||_{𝕃_{p}(T)}^{p} + ||g_{x}||_{𝕃_{p}(T,l_{2})}^{p} + ||g||_{𝕃_{p}(T,l_{2})}^{p}).

Let us take a constant c > 0 and consider (c^{2}f, cg)(c^{2}t, cx), c^{−1}w_{c^{2}t}, and u(c^{2}t, cx) instead of f, g, w, and u. Then, from the last estimate, we get

E sup_{t≤T}{c^{p−d}||u_{x}(c^{2}t, ·)||_{p}^{p} + c^{−d}||u(c^{2}t, ·)||_{p}^{p}}

≤ Ne^{T} (c^{2p−(d+2)}||f||_{𝕃_{p}(c^{2}T)}^{p} + c^{2p−(d+2)}||g_{x}||_{𝕃_{p}(c^{2}T,l_{2})}^{p} + c^{p−(d+2)}||g||_{𝕃_{p}(c^{2}T,l_{2})}^{p}),

the constant N being the same as above. It follows that, for c, T ≥ 0,

E sup_{t≤T}{||u_{x}(t, ·)||_{p}^{p} + c^{−p}||u(t, ·)||_{p}^{p}}

≤ Ne^{T/c^{2}}c^{p−2}(||f||_{𝕃_{p}(T)}^{p} + ||g_{x}||_{𝕃_{p}(T,l_{2})}^{p} + c^{−p}||g||_{𝕃_{p}(T,l_{2})}^{p}).

Upon setting c^{2} = T and considering (1 − Δ)^{(n−1)/2}u instead of u, we conclude that, under the conditions of Theorem 4.2, for any T > 0,

E{sup_{t≤T} ||u_{x}(t, ·)||_{n−1,p}^{p} + T^{−p/2} sup_{t≤T} ||u(s, ·)||_{n−1,p}^{p}} ≤

N(d, p, δ, K)T^{(p−2)/2}(||f||_{ℍ_{p}^{n−1}(T)}^{p} + ||g_{x}||_{ℍ_{p}^{n−1}(T,l_{2})}^{p} + T^{−p/2}||g||_{ℍ_{p}^{n−1}(T,l_{2})}^{p}).

We will later prove a much deeper estimate than (4.26).

5. Equations with Variable Coefficients

Here we are going to state main results concerning L_{p}–theory of SPDEs in the whole space, the most important of which, Theorem 5.1, will be proved in Sec. 6. Take a stopping time τ ≤ T with T being a finite constant. Fix n ∈ (−∞, ∞) and fix a number γ ∈ [0, 1) such that γ = 0 if n = 0, ±1, ±2, ...; otherwise γ > 0 and is such that |n| + γ is not an integer. Define

B^{|n|+γ} = \{ \begin{matrix}B(ℝ^{d}) & if n = 0, \\ C^{|n|−1,1}(ℝ^{d}) & if n = ±1, ±2, ..., \\ C^{|n|+γ}(ℝ^{d}) & otherwise; \end{matrix}

B^{|n|+γ}(l_{2}) = \{ \begin{matrix}B(ℝ^{d}, l_{2}) & if n = 0, \\ C^{|n|−1,1}(ℝ^{d}, l_{2}) & if n = ±1, ±2, ..., \\ C^{|n|+γ}(ℝ^{d}, l_{2}) & otherwise, \end{matrix}

where B(ℝ^{d}) is the Banach space of bounded functions on ℝ^{d}, C^{|n|−1,1}(ℝ^{d}) is the Banach space of |n| − 1 times continuously differentiable functions whose derivatives of (|n| − 1)st order satisfy the Lipschitz condition on ℝ^{d}, C^{|n|+γ}(ℝ^{d}) is the usual Hölder space, and l_{2} means that instead of real–valued functions we consider l_{2}–valued ones.

Consider the following nonlinear equation on [0, τ]:

\begin{matrix}du(t, x) = & [a^{ij}(t, x)u_{x^{i}x^{j}}(t, x) + f(u, t, x)] dt \\ + [σ^{ik}(t, x)u_{x^{i}}(t, x) + g^{k}(u, t, x)] dw_{t}^{k}, \end{matrix}  (5.1)

where a^{ij} and f are real-valued, and σ^{i} and g are l_{2}-valued functions defined for ω ∈ Ω, t ≥ 0, x ∈ ℝ^{d}, u ∈ H_{p}^{n+2}, i, j = 1, ..., d. We consider this equation in the sense of Definition 3.5 (where we take n + 2 instead of n).

We make the following assumptions, where, as in Section 4, we define

α^{ij}(t, x) = \frac{1}{2}(σ^{i}(t, x), σ^{j}(t, x))_{l_{2}}.  (5.2)

Assumption 5.1 (coercivity). For any ω ∈ Ω, t ≥ 0, x, λ ∈ ℝ^{d}, we have

K|λ|^{2} ≥ [a^{ij}(t, x) − α^{ij}(t, x)]λ^{i}λ^{j} ≥ δ|λ|^{2},

where K, δ are fixed strictly positive constants.

Assumption 5.2 (uniform continuity of a and σ). For any ε > 0, i, j, there exists a κ_{ε} > 0 such that

|a^{ij}(t, x) − a^{ij}(t, y)| + |σ^{i}(t, x) − σ^{i}(t, y)|_{l_{2}}^{2} ≤ ε  (5.3)

whenever |x − y| ≤ κ_{ε}, t ≥ 0, ω ∈ Ω.

This assumption is actually used only if n = 0, and even then we need a stronger condition on σ. For other values of n we impose stronger conditions on a and σ.

Assumption 5.3. For any i, j, k, the functions a^{ij}(t, x) and σ^{ik}(t, x) are real-valued P × B(ℝ^{d})–measurable functions, and for any ω ∈ Ω and t ≥ 0, we have

a^{ij}(t, ·) ∈ B^{|n|+γ}, σ^{i}(t, ·) ∈ B^{|n+1|+γ}(l_{2}).

Assumption 5.4. For any u ∈ H_{p}^{n+2}, the functions f(u, t, x) and g(u, t, x) are predictable as functions taking values in H_{p}^{n} and H_{p}^{n+1}(ℝ^{d}, l_{2}), respectively.

Assumption 5.5. For any t ≥ 0, ω, i, j,

||a^{ij}(t, ·)||_{B^{|n|+γ}} + ||σ^{i}(t, ·)||_{B^{|n+1|+γ}(l_{2})} ≤ K, (f(0, ·, ·), g(0, ·, ·)) ∈ F_{p}^{n}(τ).

Assumption 5.6. The functions f, g are continuous in u. Moreover, for any ε > 0, there exists a constant K_{ε} such that, for any u, v ∈ H_{p}^{n+2}, t, ω, we have

||f(u, t, ·) − f(v, t, ·)||_{n,p} + ||g(u, t, ·) − g(v, t, ·)||_{n+1,p}

≤ ε||u − v||_{n+2,p} + K_{ε}||u − v||_{n,p}. (5.4)

Theorem 5.1. Let Assumptions 5.1–5.6 be satisfied and let

u_{0} ∈ L_{p}(Ω, F_{0}, H_{p}^{n+2−2/p}).

Then the Cauchy problem for equation (5.1) on [0, τ] with the initial condition u(0, ·) = u_{0} has a unique solution u ∈ H_{p}^{n+2}(τ). For this solution, we have

||u||_{H_{p}^{n+2}(τ)} ≤ N{||f(0, ·, ·)||_{ℍ_{p}^{n}(τ)} + ||g(0, ·, ·)||_{ℍ_{p}^{n+1}(τ,l_{2})} + (E||u_{0}||_{n+2−2/p,p}^{p})^{1/p}},

where the constant N depends only on d, n, γ, p, δ, K, T, and the functions κ_{ε} and K_{ε}.

To discuss the theorem, we need the following lemma.

Lemma 5.2. Let ζ ∈ C_{0}^{∞} be a nonnegative function such that ∫ ζ(x) dx = 1 and define ζ_{k}(x) = k^{d}ζ(kx), k = 1, 2, 3, .... We assert that, for any u ∈ H_{p}^{n}, we have the following:

(i) ||au||_{n,p} ≤ Nā||u||_{n,p}, where ā = ||a||_{B^{|n|+γ}} and the constant N depends only on d, p, n, and γ;

(ii) ||u ∗ ζ_{k}||_{n,p} ≤ ||u||_{n,p}, ||u − u ∗ ζ_{k}||_{n,p} → 0.

The same assertions hold true for Banach–space valued a with natural definition of ā.

Proof. If n ≠ 0, ±1, ±2, ... (and γ > 0), then one gets (i) by Corollary 2.8.2 (ii) of [33]. If n is a nonnegative integer, then (i) follows from the Leibnitz rule (and the fact that H_{p}^{n} = W_{p}^{n}). For negative integers (and generally for negative n) (i) follows easily by duality, that is, by using the fact that if u = (1 − Δ)^{−n/2}f, then

(au, ϕ) = (f, (1 − Δ)^{−n/2}(aϕ)) ≤ ||f||_{p} ||aϕ||_{−n,q} q = p/(p − 1).

As for (ii), the first inequality follows from Minkowski’s inequality and the second one is derived as usual owing to denseness of C_{0}^{∞} in H_{p}^{n}. The lemma is proved.

Remark 5.3. As we have said above, by solution to the Cauchy problem for equation (5.1) on [0, τ] with the given initial condition u_{0}, we understand a function u ∈ H_{p}^{n+2}(τ) such that for any test function ϕ ∈ C_{0}^{∞}, one has almost surely

\begin{aligned}(u(t, ·), ϕ) = (u_{0}, ϕ) & + ∫_{0}^{t}(a^{ij}(s, ·)u_{x^{i}x^{j}}(s, ·) + f(u, s, ·), ϕ) ds \\ & + ∫_{0}^{t}(σ^{ik}(s, ·)u_{x^{i}}(s, ·) + g^{k}(u, s, ·), ϕ) dw_{s}^{k}, ∀t ∈ [0, τ]. \end{aligned}

It is important that, under our assumptions, the equation makes sense for u ∈ H_{p}^{n+2}(τ), since by Lemma 5.2 we have a^{ij}u_{x^{i}x^{j}} ∈ H_{p}^{n}, σ^{i}u_{x^{i}} ∈ H_{p}^{n+1}(ℝ^{d}, l_{2}) whenever u ∈ H_{p}^{n+2}.

Remark 5.4. Two main ideas in the proof of this theorem are quite standard. The first one, reduction to equations with constant coefficients, will be seen very clearly. The second one, which is somewhat hidden, consists of introducing the new unknown function v = (1 − Δ)^{n/2}u, which reduces the case of general n to the case n = 0. The function v satisfies

dv = {(1 − Δ)^{n/2}(a^{ij}(1 − Δ)^{−n/2}v_{x^{i}x^{j}}) + (1 − Δ)^{n/2}f} dt
+ {(1 − Δ)^{n/2}(σ^{ik}(1 − Δ)^{−n/2}v_{x^{i}}) + (1 − Δ)^{n/2}g^{k}} dw_{t}^{k}.

This is a pseudo–differential equation, and we note that more general pseudo–differential equations can be considered too. Also, this equations shows a need to have smoothness assumptions on a, σ in x if we are interested in n ≠ 0 both positive or negative.

Remark 5.5. By Theorem 14.2 of [14], for any u ∈ H_{p}^{n+2} and m ∈ [n, n + 2], we have

||u||_{m,p} ≤ N||u||_{n+2,p}^{θ}||u||_{n,p}^{1−θ} ≤ Nθε||u||_{n+2,p} + N(1 − θ)ε^{−θ/(1−θ)}||u||_{n,p},

where θ = (m−n)/2 and N depends only on d, n, m, and p. This shows that the right–hand side in (5.4) can be replaced by ||u − v||_{n+ε+1,p} once |ε| < 1. As an example, one can take f = f_{0}(x) sup_{x} |u_{x}| if (n + 1)p > d and f_{0} ∈ H_{p}^{n}. Indeed, by Sobolev’s embedding theorems, H_{p}^{n+1+ε} ⊂ C^{1} if (n + ε)p > d. Therefore,

||f(u, t, ·) − f(v, t, ·)||_{n,p} ≤ ||f_{0}||_{n,p} sup_{x} |(u − v)_{x}| ≤ N||u − v||_{n+1+ε,p}.

Remark 5.6. A typical application of Theorem 5.1 occurs when f(u, t, x) = b^{i}(t, x)u_{x^{i}}+ c(t, x)u + f(t, x) and g(u, t, x) = ν(t, x)u + g(t, x), so that (5.1) becomes

du = (a^{ij}u_{x^{i}x^{j}} + b^{i}u_{x^{i}} + cu + f) dt + (σ^{ik}u_{x^{i}} + ν^{k}u + g^{k}) dw_{t}^{k}.  (5.5)

To describe the appropriate assumptions, we take ε ∈ (0, 1) and denote

n_{b} = n + γ  if n ≥ 0, n_{b} = 0 if n ∈ (−1, 0],
n_{ν} = n + 1 + γ if n ≥ −1, n_{ν} = 0 if n ∈ (−2, −1],
n_{c} = n + γ  if n ≥ 0, n_{c} = 0 if n ∈ (−2, 0],

\begin{matrix}n_{b} = −n − 1 + ε & if & n ≤ −1, \\ n_{ν} = −n − 2 + ε & if & n ≤ −2, \\ n_{c} = −n − 2 + ε & if & n ≤ −2. \end{matrix}

Assume that b, c, and ν are appropriately measurable and

b^{i}(t, ·) ∈ B^{n_{b}}, c(t, ·) ∈ B^{n_{c}}, ν(t, ·) ∈ B^{n_{ν}} (ℝ^{d}, l_{2}),
f(t, ·) ∈ H_{p}^{n}, g(t, ·) ∈ H_{p}^{n+1}(ℝ^{d}, l_{2}).

||b^{i}(t, ·)||_{B^{n_{b}}} + ||c(t, ·)||_{B^{nc}} + ||ν(t, ·)||_{B^{nν} (ℝ^{d},l_{2})} ≤ K, (f(·, ·), g(·, ·)) ∈ F_{p}^{n}(τ). It turns out then that the assumptions of Theorem 5.1 about f(u, t, x) and g(u, t, x) are satisfied. To show this, it suffices to apply Remark 5.5 and to notice that, for instance, if n ≥ −1, then ||νu||_{n+1,p} ≤ N||u||_{n+1,p} by Lemma 5.2; if n ∈ (−2, −1], then obviously ||νu||_{n+1,p} ≤ ||νu||_{p} ≤ N||u||_{p} = N||u||_{n+1+(−n−1),p}, and −n − 1 ∈ [0, 1); if n ≤ −2, then Lemma 5.2 yields ||νu||_{n+1,p} ≤ ||νu||_{n+2−ε_{1},p} ≤ N||u||_{n+2−ε_{1},p}, where ε_{1} ∈ (0, ε). The terms ||b^{i}u_{x^{i}}||_{n,p}, ||cu||_{n,p} are considered similarly.

Actually, the above conditions on b, c, and ν can be considerably relaxed if one makes use of deeper theorems about multipliers from [33].

Conditions (5.6) and (5.7) of the following theorem are discussed in Remarks 5.9 and 5.8.

Theorem 5.7. Assume that for m = 1, 2, 3, ..., we are given a_{m}^{ij}, σ_{m}^{i}, f_{m}, g_{m}, and u_{0m} having the same sense as in Theorem 5.1 and verifying the same assumptions as a^{ij}, σ^{i}, f, g, and u_{0} with the same constants δ, K, κ_{ε}, and K_{ε}. Let ζ(x) be a real function of class C_{0}^{∞} such that ζ(x) = 1 if |x| ≤ 1 and ζ(x) = 0 if |x| ≥ 2. Define ζ_{r}(x) = ζ(x/r) and assume that, for any r = 1, 2, 3, ..., i, j = 1, ..., d, t ≥ 0, and ω ∈ Ω,

||ζ_{r}{a^{ij}(t, ·) − a_{m}^{ij}(t, ·)}||_{n,p} + ||ζ_{r}{σ^{i}(t, ·) − σ_{m}^{i}(t, ·)}||_{n+1,p} → 0  (5.6)

as m → ∞. Finally, let E||u_{0m} − u_{0}||_{n+2−2/p,p}^{p} → 0 and

||f(u, ·, ·) − f_{m}(u, ·, ·)||_{ℍ_{p}^{n}(τ)} → 0, ||g(u, ·, ·) − g_{m}(u, ·, ·)||_{ℍ_{p}^{n+1}(τ,l_{2})} → 0  (5.7)

whenever u ∈ H_{p}^{n+2}(τ). Take the function u from Theorem 5.1 and for any m define u_{m} ∈ H_{p}^{n+2}(τ) as the (unique) solution of the Cauchy problem for the equation

\begin{matrix}du_{m}(t, x) = & [a_{m}^{ij}(t, x)u_{mx^{i}x^{j}}(t, x) + f_{m}(u_{m}, t, x)] dt \\ + [σ_{m}^{ik}(t, x)u_{mx^{i}}(t, x) + g_{m}^{k}(u_{m}, t, x)] dw_{t}^{k} \end{matrix}  (5.8)

on [0, τ] with initial condition u_{m}(0, ·) = u_{0m}. Then ||u − u_{m}||_{H_{p}^{n+2}(τ)} → 0 as m → ∞.

Proof. For v_{m} = u − u_{m} we have
dv_{m}(t) = [a_{m}^{ij}v_{mx^{i}x^{j}} + F_{m}(v_{m})] dt + [σ_{m}^{ik}v_{mx^{i}} + G_{m}^{k}(v_{m})] dw_{t}^{k},

where

F_{m}(v) = (a^{ij} − a_{m}^{ij})u_{x^{i}x^{j}} + f(u) − f_{m}(u − v),
G_{m}^{k}(v) = (σ^{ik} − σ_{m}^{ik})u_{x^{i}} + g^{k}(u) − g_{m}^{k}(u − v).

Hence, from our assumptions and by Theorem 5.1, we obtain

||u − u_{m}||_{H_{p}^{n+2}(τ)} ≤ NI_{m},

where N is independent of m and

I_{m} = ||(a^{ij} − a_{m}^{ij})u_{x^{i}x^{j}}||_{ℍ_{p}^{n}(τ)} + ||f(u) − f_{m}(u)||_{ℍ_{p}^{n}(τ)}

+||(σ^{i} − σ_{m}^{i})u_{x^{i}}||_{ℍ_{p}^{n+1}(τ,l_{2})} + ||g(u) − g_{m}(u)||_{ℍ_{p}^{n+1}(τ,l_{2})} + (E||u_{0} − u_{0m}||_{n+2−2/p,p}^{p})^{1/p}.

Next, by our assumptions about convergence of f_{m}, g_{m}, u_{0m},

lim sup_{m→∞} I_{m} ≤ lim sup_{m→∞}{||(a^{ij} − a_{m}^{ij})u_{x^{i}x^{j}}||_{ℍ_{p}^{n}(τ)} + ||(σ^{i} − σ_{m}^{i})u_{x^{i}}||_{ℍ_{p}^{n+1}(τ,l_{2})}}.  (5.9)
Here, by Lemma 5.2, for any v ∈ C_{0}^{∞} and r so large that vζ_{r} = v, we have
||(a^{ij} − a_{m}^{ij})u_{x^{i}x^{j}}||_{n,p} ≤ N||(u − v)_{x^{i}x^{j}}||_{n,p} + ||(a^{ij} − a_{m}^{ij})v_{x^{i}x^{j}}||_{n,p},  (5.10)
||(a^{ij} − a_{m}^{ij})v_{x^{i}x^{j}}||_{n,p} = ||ζ_{r}(a^{ij} − a_{m}^{ij})v_{x^{i}x^{j}}||_{n,p} ≤ N||ζ_{r}(a^{ij} − a_{m}^{ij})||_{n,p}||v||_{B^{|n|+2+γ}} ,

where the constants N do not depend on m and r. Thus,

lim sup_{m→∞} ||(a^{ij} − a_{m}^{ij})u_{x^{i}x^{j}}||_{n,p} ≤ N||(u − v)_{x^{i}x^{j}}||_{n,p},

and, from the arbitrariness of v, we conclude that the left-hand side is zero for those ω and t for which u ∈ H_{p}^{n+2}. If we again apply Lemma 5.2, then we see that the pth power of the left-hand side of (5.10) is bounded by an integrable function. This and the dominated convergence theorem imply that

lim_{m→∞} ||(a^{ij} − a_{m}^{ij})u_{x^{i}x^{j}}||_{ℍ_{p}^{n}(τ)} = 0.

Similar arguments take care of the remaining term in (5.9). The theorem is proved.

Remark 5.8. Condition (5.7) is satisfied for any u ∈ H_{p}^{n+2}(τ) if and only if it is satisfied for u(t, x) ≡ ϕ(x) with any ϕ ∈ C_{0}^{∞}.

Indeed, the “only if” part is obvious. In the proof of “if” part notice that, under the “if” assumption, (5.7) is automatically satisfied for u of type

v = ∑_{i=1}^{j} I_{(τ_{i−1},τ_{i}]}(t)v_{i}(x),

where τ_{i} are bounded stopping times and v_{i} ∈ C_{0}^{∞}. By Theorem 3.10, one can approximate any u ∈ H_{p}^{n+2}(τ) ⊂ ℍ_{p}^{n+2}(τ) with functions like v. It remains only to notice that, by the assumptions of the theorem, for instance,

||f(u, ·, ·) − f_{m}(u, ·, ·)||_{ℍ_{p}^{n}(τ)} ≤ ||f(u, ·, ·) − f(v, ·, ·)||_{ℍ_{p}^{n}(τ)} + ||f(v, ·, ·) − f_{m}(v, ·, ·)||_{ℍ_{p}^{n}(τ)}

+||f_{m}(v, ·, ·) − f_{m}(u, ·, ·)||_{ℍ_{p}^{n}(τ)} ≤ N||u − v||_{ℍ_{p}^{n+2}(τ)} + ||f(v, ·, ·) − f_{m}(v, ·, ·)||_{ℍ_{p}^{n}(τ)}.

Remark 5.9. While checking conditions (5.6) and (5.7), it is useful to bear in mind that, if one defines σ_{m}^{ik} = σ^{ik} and g_{m}^{k} = g^{k} for k ≤ m and σ_{m}^{ik} = g_{m}^{k} = 0 for k > m, then

||ζ_{r}{σ^{i}(t, ·) − σ_{m}^{i}(t, ·)}||_{n+1,p} → 0, ||g(u, ·, ·) − g_{m}(u, ·, ·)||_{ℍ_{p}^{n+1}(τ,l_{2})} → 0.

Indeed, for instance,

||g(u, ·, ·) − g_{m}(u, ·, ·)||_{ℍ_{p}^{n+1}(τ,l_{2})}^{p} = E ∫_{0}^{τ} ||( ∑_{k>m} |(1 − Δ)^{(n+1)/2}g^{k}(u, t, ·)|^{2})^{1/2}||_{p}^{p} dt,

which goes to zero by the dominated convergence theorem.

This fact allows one to approximate solutions of (5.1) by solutions of

\begin{matrix}du_{m}(t, x) = & [a^{ij}(t, x)u_{mx^{i}x^{j}}(t, x) + f(u_{m}, t, x)] dt \\ + ∑_{k≤m}[σ^{ik}(t, x)u_{mx^{i}}(t, x) + g^{k}(u_{m}, t, x)] dw_{t}^{k}. \end{matrix}  (5.11)

Before stating the following corollary, take the functions ζ_{k} from Lemma 5.2 and, for a function h = h(u, t, x) defined for t > 0, x ∈ ℝ^{d}, and u in a function space, let

ĥ_{m}(u, t, x) = ∫_{ℝ^{d}} h(u(· + y), t, x − y)ζ_{m}(y) dy.  (5.12)

To get a better idea about this definition, notice that if, for instance, h(u, t, x) = c(t, x)u(t, x), where c(t, x) is a given function, then h(u(· + y), t, x − y) = c(t, x − y)u(t, x), so that ĥ_{m}(u, t, x) = c_{m}(t, x)u(t, x) with c_{m}(t, x) = c(t, x) ∗ ζ_{m}(x).

Corollary 5.10. Under the assumptions of Theorem 5.1, define

(a_{m}, σ_{m})(t, x) = (a, σ)(t, x) ∗ ζ_{m}(x),
(f_{m}, g_{m}) = (f̂_{m}, ĝ_{m}) or (f_{m}, g_{m})(u, t, x) = (f, g)(u, t, x) ∗ ζ_{m}(x).

Then the assumptions of Theorem 5.7 are satisfied, and if we define u_{m} ∈ H_{p}^{n+2}(τ) as a solution of the Cauchy problem (5.8) with initial condition u_{m}(0, ·) = u_{0} ∗ ζ_{m}, then ||u − u_{m}||_{H_{p}^{n+2}(τ)} → 0.

To prove this, notice that

(σ_{m}^{i}, σ_{m}^{j})_{l_{2}}λ^{i}λ^{j} = |(σ^{i}λ^{i}) ∗ ζ_{m}|_{l_{2}}^{2} ≤ ζ_{m} ∗ [(σ^{i}, σ^{j})_{l_{2}}λ^{i}λ^{j}],

which guarantees that the approximating equations satisfy the same coercivity condition. Also, for any u ∈ H_{p}^{n+2} and ϕ ∈ C_{0}^{∞}, the function (f(u(· + z), t, · − y), ϕ) = (f(u(· + z), t, ·), ϕ(· + y)) is continuous in z owing to Assumption 5.6 and infinitely differentiable in y as it has to be for any distribution. Therefore, the definition of f̂_{m} according to (5.12) makes sense as an integral of distribution with respect to the parameter y, namely

(f̂_{m}(u, t, ·), ϕ) := ∫_{ℝ^{d}} ζ_{m}(y)(f(u(· + y), t, · − y), ϕ) dy.

Furthermore, f̂_{m} = f̂_{0m} + f̂_{1m}, where f_{0} = f(0, t, x) and f_{1} = f(u, t, x) − f(0, t, x). By Lemma 5.2, we have f̂_{0m}(t, ·) ∈ H_{p}^{n}. By Assumption 5.6,

|(f̂_{1m}(u, t, ·), ϕ)| ≤ N ∫_{ℝ^{d}} ζ_{m}(y)||ϕ||_{−n,q}||u(· + y)||_{n+2,p} dy = N||ϕ||_{−n,q}||u||_{n+2,p},

where q = p/(p − 1). Hence, f̂_{m} is a H_{p}^{n}–valued function. Also, f(u, t, x) ∗ ζ_{m}(x) is well defined because of Assumption 5.4. Similarly, g_{m} is well defined. Minkowski’s inequality implies that all Assumptions 5.1–5.6 are satisfied for any m with the same K, δ, κ_{ε}, K_{ε}. Other assumptions of Theorem 5.7 are satisfied due to continuity in the mean of summable functions and Assumptions 5.1–5.6. For instance, for any ϕ ∈ C_{0}^{∞},

||f(ϕ, ·, ·) − f̂_{m}(ϕ, ·, ·)||_{ℍ_{p}^{n}(τ)} ≤ I_{1m} + I_{2m},

where

I_{1m} := ||f(ϕ, ·, ·) − f_{m}(ϕ, ·, ·)||_{ℍ_{p}^{n}(τ)}, f_{m}(ϕ, t, x) = ∫_{ℝ^{d}} f(ϕ, t, x − y)ζ_{m}(y) dy,
I_{2m} := ||f_{m}(ϕ, ·, ·) − f̂_{m}(ϕ, ·, ·)||_{ℍ_{p}^{n}(τ)}
≤ ∫_{ℝ^{d}} ||f(ϕ, ·, · − y) − f(ϕ(· + y), ·, · − y)||_{ℍ_{p}^{n}(τ)}ζ_{m}(y) dy

= ∫_{ℝ^{d}} ||f(ϕ, ·, ·) − f(ϕ(· + y), ·, ·)||_{ℍ_{p}^{n}(τ)}ζ_{m}(y) dy ≤ N ∫_{ℝ^{d}} ||ϕ − ϕ(· + y)||_{n+2,p}ζ_{m}(y) dy.

Clearly (cf. Lemma 5.2) I_{1m} + I_{2m} → 0 as m → ∞, and the application of Remark 5.8 ends the argument.

Corollary 5.11. Let the assumptions of Theorem 5.1 be satisfied and also let them be satisfied for a p = q, where q ≥ 2. Then the solution u from Theorem 5.1 belongs also to H_{q}^{n+2}(τ).

To prove this, without loss of generality, assume p < q and let v be the solution of the same initial value problem but belonging to H_{q}^{n+2}(τ) (such a unique v exists by Theorem 5.1). We have only to show that v = u. In light of Corollary 5.10, we can approximate both v and u by solutions of equations with smooth coefficients and with (f_{m}, g_{m})(u, t, x) = (f, g)(u, t, x) ∗ ζ_{m}(x). We only need to show that the approximating solutions coincide. Observe that, for any r ≥ n, for instance,

||f_{m}(u, ·, ·) − f_{m}(v, ·, ·)||_{r,p} ≤ N||f(u, ·, ·) − f(v, ·, ·)||_{n,p} ≤ Nε||u − v||_{n+2,p}
+NK_{ε}||u − v||_{n,p} ≤ Nε||u − v||_{r+2,p} + NK_{ε}||u − v||_{r,p},

where N’s depend only on m, r, n, p. This shows that, for any fixed m, Assumptions 5.1–5.6 are satisfied with any large n both for p and q.

Therefore, we can suppose from the very beginning that Assumptions 5.1–5.6 are satisfied for any large n for p and q. In this case, by Theorem 5.1, u ∈ H_{p}^{r}(τ) for any r, and hence, also invoking Theorem 3.7,

E ∫_{0}^{τ} ||u_{xx}(t, ·)||_{r,p}^{p} dt < ∞, E sup_{t≤τ} ||u(t, ·)||_{C^{r}}^{p} < ∞

for any r = 1, 2, 3, .... Since ||u_{xx}(t, ·)||_{r,i}^{i} ≤ N||u_{xx}(t, ·)||_{r,p}^{p}||u(t, ·)||_{C^{r+2}}^{i−p} for i ≥ p, it follows that

∫_{0}^{τ} ||u_{xx}(t, ·)||_{r,i}^{i} dt < ∞ (a.s.)

for any i ≥ p. Take here r = 0 and i = q and define

τ_{k} = τ ∧ inf{t : ∫_{0}^{t} ||u_{xx}(s, ·)||_{q}^{q} ds ≥ k}.

Then obviously u ∈ H_{q}^{2}(τ_{k}). Since v lies in the same class, by uniqueness u(t, ·) = v(t, ·) for t ≤ τ_{k} (a.s.). It remains only to observe that τ_{k} ↑ τ when k → ∞.

Next general result concerns the maximum principle.

Theorem 5.12 (maximum principle). Let the assumptions of Theorem 5.1 be satisfied and let u be the function existence of which is asserted in this theorem. Assume that

f(v, t, x) = b^{i}(t, x)v_{x^{i}}(t, x) + v(t, x)c(t, x) + f(t, x), g^{k}(v, t, x) = v(t, x)ν^{k}(t, x),

where b^{i}(t, x), c(t, x), ν^{k}(t, x) are certain bounded functions on ^{|}(0, τ]] × ℝ^{d} and f(t, x) = f(0, t, x) ≥ 0. Also assume that for any ω we have u_{0} ≥ 0. Then u(t, ·) ≥ 0 for all t ∈ [0, τ] almost surely.

Proof. By virtue of Remark 5.9, we can only concentrate on equations with finite number of Wiener processes like (5.11), which in our case is the following

du(t, x) = [a^{ij}(t, x)u_{x^{i}x^{j}}(t, x) + b^{i}(t, x)u_{x^{i}}(t, x) + u(t, x)c(t, x) + f(t, x)] dt

+ ∑_{k≤m}[σ^{ik}(t, x)u_{x^{i}}(t, x) + u(t, x)ν^{k}(t, x)] dw_{t}^{k}. (5.13)

Corollary 5.10 and the explanation after (5.12) allow us to assume that u_{0}(x) and the coefficients and f in (5.13) are infinitely differentiable in x. After this, by multiplying u_{0} and f by a cut–off function of x and by using Remark 5.6 and Theorem 5.7, we convince ourselves that we can assume that u_{0} and f(t, x) have supports with respect to x in a fixed ball. In this case the assumptions of Theorem 5.1 are satisfied for p = 2 again by Remark 5.6 (and Hölder’s inequality), which, by Corollary 5.11, yields u ∈ H_{2}^{n}(τ) for any n. Now our assertion follows from the maximum principle proved in [25] (see Theorem 4.2 there). The only point to mention is that in [25] we considered (5.13) on [0, T], but we always can continue our data in an appropriate way after τ, which is assumed to be less than T. The theorem is proved.

By the way the proof of the maximum principle in [25] is based on representation formulas like (4.28). A heuristic derivation of many other formulas of that kind can be found in [15].

6. Proof of Theorem 5.1

The proof we present here is quite typical for proofs of solvability of equations with variable coefficients on the basis of solvability of equations with constant ones. The same type of arguments is commonly used in the theory of partial differential equations for proving the solvability in Sobolev or Hölder spaces. First we need some auxiliary constructions. Fix a T ∈ (0, ∞).

Definition 6.1. Assume that, for ω ∈ Ω and t ≥ 0, we are given operators

L(t, ·) : H_{p}^{n+2} → H_{p}^{n}, Λ(t, ·) : H_{p}^{n+2} → H_{p}^{n+1}(ℝ^{d}, l_{2}).

Assume that (i) for any ω and t, the operators L(t, u) and Λ(t, u) are continuous (with respect to u); (ii) for any u ∈ H_{p}^{n+2}, the processes L(t, u) and Λ(t, u) are predictable; (iii) for any ω ∈ Ω, t ≥ 0, and u ∈ H_{p}^{n+2}, we have

||L(t, u)||_{n,p} + ||Λ(t, u)||_{n+1,p} ≤ N_{L,Λ}(1 + ||u||_{n+2,p}),

where N_{L,Λ} is a constant.

Then for a function u ∈ H_{p}^{n+2}(T), we write

(L, Λ)u = −(f, g)

if (f, g) ∈ F_{p}^{n}(T), and, in the sense of Definition 3.5, for t ∈ [0, T], we have that 𝔻u(t) = L(t, u(t)) + f(t) and 𝕊u(t) = Λ(t, u(t)) + g(t), or put otherwise

u(t) = u(0) + ∫_{0}^{t}(L(s, u(s)) + f(s)) ds + ∫_{0}^{t}(Λ^{k}(s, u(s)) + g^{k}(s)) dw_{s}^{k} (a.s.).

Remark 6.2. By virtue of our conditions on L and Λ, for any u ∈ H_{p}^{n+2}(T), we have (L(u), Λ(u)) ∈ F_{p}^{n}(T). Also, (L, Λ)u = (L(u) − 𝔻u, Λ(u) − 𝕊u). In particular, the operator (L, Λ) is well defined on H_{p}^{n+2}(T), and, as follows easily from Definition 6.1 (iii),

||(L, Λ)u||_{F_{p}^{n}(T)} ≤ (1 + 2N_{L,Λ})||u||_{H_{p}^{n+2}(T)} + 2N_{L,Λ}T^{1/p}.

In terms of Definition 6.1, Theorem 4.10 has the following version.

Theorem 6.3. Let a and σ satisfy the assumptions from the beginning of Sec. 4. Define

Lu = a^{ij}u_{x^{i}x^{j}}, Λu = σ^{i}u_{x^{i}}.

Then the operator (L, Λ) is a one-to-one operator from H_{p,0}^{n+2}(T) onto F_{p}^{n}(T) and the norm of its inverse is less than a constant depending only on d, p, δ, and K (thus independent of T).

Next, we prove a perturbation result. It needs a proof because we do not allow ε to depend on T.

Theorem 6.4. Take the operators L and Λ from Theorem 6.3, and let some operators L_{1} and Λ_{1} satisfy the requirements from Definition 6.1. We assert that there exists a constant ε ∈ (0, 1) depending only on d, p, δ, and K such that if, for a constant K_{1} and any u, v ∈ H_{p}^{n+2}, t ≥ 0, ω ∈ Ω, we have

||L_{1}(t, u) − L_{1}(t, v)||_{n,p} + ||Λ_{1}(t, u) − Λ_{1}(t, v)||_{n+1,p}
≤ ε||u_{xx} − v_{xx}||_{n,p} + K_{1}||u − v||_{n+1,p},  (6.1)

then, for any (f, g) ∈ F_{p}^{n}(T), there exists a unique solution u ∈ H_{p,0}^{n+2}(T) of the equation

(L + L_{1}, Λ + Λ_{1})u = −(f, g).  (6.2)

Furthermore, for this solution u, we have

||u||_{H_{p}^{n+2}(T)} ≤ N||(L_{1}(·, 0) + f, Λ_{1}(·, 0) + g)||_{F_{p}^{n}(T)},  (6.3)

where N depends only on d, p, δ, K, K_{1}, and T and N is independent of T if K_{1} = 0.

Proof. First notice that, by interpolation theorems, ||u||_{n+1,p} ≤ ε||u_{xx}||_{n,p}+N(ε, d, p)||u||_{n,p}. Therefore, without loss of generality we assume that instead of (6.1) we have

||L_{1}(t, u) − L_{1}(t, v)||_{n,p} + ||Λ_{1}(t, u) − Λ_{1}(t, v)||_{n+1,p}
≤ ε||u_{xx} − v_{xx}||_{n,p} + K_{1}||u − v||_{n,p}.

Now fix (f, g) ∈ F_{p}^{n}(T). Take u ∈ H_{p,0}^{n+2}(T), observe that (L_{1}(u), Λ_{1}(u)) ∈ F_{p}^{n}(T), and, by using Theorem 6.3, define v ∈ H_{p,0}^{n+2}(T) as the unique solution of the equation (L, Λ)v = −(f + L_{1}(u), g + Λ_{1}(u)). By denoting v = Ru, we define an operator R : H_{p,0}^{n+2}(T) → H_{p,0}^{n+2}(T). Equation (6.2) is equivalent to the equation u = Ru. Therefore, to prove the existence and uniqueness of solutions to (6.2), it suffices to show that, for an integer m > 0, the operator R^{m} is a contraction in H_{p,0}^{n+2}(T).

By Theorem 6.3, for t ≤ T,

||Ru − Rv||_{H_{p}^{n+2}(t)}^{p} ≤ N||(L_{1}(u) − L_{1}(v), Λ_{1}(u) − Λ_{1}(v))||_{F_{p}^{n}(t)}^{p}
≤ N_{0}ε^{p}||u − v||_{H_{p}^{n+2}(t)}^{p} + N_{0}K_{1}^{p} ∫_{0}^{t} E||u(s) − v(s)||_{n,p}^{p} ds,

with a constant N_{0} depending only on d, p, δ, and K. This gives the desired result if K_{1} = 0. Also, in this case estimate (6.3) follows obviously with N independent of T.

In the general case, by Theorem 3.7,

E||u(s) − v(s)||_{n,p}^{p} ≤ N_{1}||u − v||_{H_{p}^{n+2}(s)}^{p},

where s ≤ T and N_{1} depends only on d, p, and T. It follows that, for t ≤ T and θ := N_{0}ε^{p}, we have

||Ru − Rv||_{H_{p}^{n+2}(t)}^{p} ≤ θ||u − v||_{H_{p}^{n+2}(t)}^{p} + N_{2} ∫_{0}^{t} ||u − v||_{H_{p}^{n+2}(s)}^{p} ds,

where N_{2} depends only on d, p, δ, K, K_{1}, and T. Hence, by induction,

||R^{m}u − R^{m}v||_{H_{p}^{n+2}(t)}^{p} ≤ θ^{m}||u − v||_{H_{p}^{n+2}(t)}^{p}

+ ∑_{k=1}^{m} \binom{m}{k}θ^{m−k}N_{2}^{k} ∫_{0}^{t} \frac{(t − s)^{k−1}}{(k − 1)!} ||u − v||_{H_{p}^{n+2}(s)}^{p} ds, ||R^{m}u − R^{m}v||_{H_{p}^{n+2}(T)}^{p}

≤ ∑_{k=0}^{m} \binom{m}{k}θ^{m−k} \frac{1}{k!}(T N_{2})^{k}||u − v||_{H_{p}^{n+2}(T)}^{p} ≤ 2^{m}θ^{m} max_{k} \frac{1}{k!}(T N_{2}/θ)^{k}||u − v||_{H_{p}^{n+2}(T)}^{p}.

This allows us to find ε depending only on d, p, δ, and K and m depending on the same things plus K_{1} and T, so that R^{m} is a contraction in H_{p,0}^{n+2}(T) with coefficient 1/2. Of course, this yields all our assertions. The theorem is proved.

We finish our preparations by showing how Lemma 5.2 will be used.

Remark 6.5. To some extent, in what follows, the most important consequence of assertion (i) in Lemma 5.2 is that if ā = ||a||_{B^{|n|+γ}} < ∞, then there exists a new norm ]| · |[_{n,p} in H_{p}^{n} such that

]|au|[_{n,p}≤ 2N||a||_{B}]|u|[_{n,p} (||a||_{B} = sup_{x∈ℝ^{d}} |a(x)|),

where N is the same constant as in Lemma 5.2. To show this, it suffices to observe that, for a_{m}(x) = a(x/m), u_{m}(x) = u(x/m), and m ≥ 1, we have

||(m^{2} − Δ)^{n/2}(au)||_{p} = m^{n−d/p}||(1 − Δ)^{n/2}(a_{m}u_{m})||_{p} ≤
Nā_{m}m^{n−d/p}||(1 − Δ)^{n/2}u_{m}||_{p} = Nā_{m}||(m^{2} − Δ)^{n/2}u||_{p} ≤

N(||a||_{B} + \frac{1}{m^{(|n|+γ)∧1}} ā)||(m^{2} − Δ)^{n/2}u||_{p}.

Alternatively, it would be sufficient for our needs to know that

||au||_{n,p} ≤ N(||a||_{B}||u||_{n,p} + ||a||_{B^{|n|+γ}} ||u||_{n−1,p}).

The author could not find the last inequality in the literature, though some very interesting information related to the subject can be found in [13].

Now we perform the main step in proving Theorem 5.1.

Lemma 6.6. Let Assumptions 5.1, 5.3, 5.4, and 5.5 be satisfied. Then there exists an ε = ε(d, p, n, γ, δ, K) > 0 such that if τ = T and

(i) inequality (5.3) holds with this ε for all x, y, t, and ω and

(ii) f and g are independent of u, then there exists a unique solution u ∈ H_{p,0}^{n+2}(T) of equation (5.1). Furthermore, for this solution u, we have

||u||_{H_{p}^{n+2}(T)} ≤ N||(f, g)||_{F_{p}^{n}(T)},

where N depends only on d, p, δ, K, and T.

Proof. Define a(t) = a(t, 0) and σ(t) = σ(t, 0), take operators L and Λ from Theorem 6.3 corresponding to these a(t) and σ(t), and let

L_{1}(t, u)(x) = [a^{ij}(t, x) − a^{ij}(t)]u_{x^{i}x^{j}}(x), Λ_{1}(t, u)(x) = [σ^{i}(t, x) − σ^{i}(t)]u_{x^{i}}(x).

In view of Theorem 6.4, to prove existence and uniqueness, we have only to check that if ε in (5.3) is sufficiently small, then the operators L_{1} and Λ_{1} satisfy condition (6.1) with as small ε as we like and with K_{1} under control. Observe that by Lemma 5.2,

||L_{1}(t, u)||_{n,p} ≤ N||a(t, ·) − a(t, 0)||_{B^{|n|+γ}} ||u_{xx}||_{n,p},
||Λ_{1}(t, u)||_{n+1,p} ≤ N||σ(t, ·) − σ(t, 0)||_{B^{|n+1|+γ}(l_{2})}||u_{x}||_{n+1,p}.

Since ||u_{x}||_{n+1,p} ≤ N(||u_{xx}||_{n,p} + ||u||_{n+1,p}), our lemma holds true if

||a(t, ·) − a(t, 0)||_{B^{|n|+γ}} + ||σ(t, ·) − σ(t, 0)||_{B^{|n+1|+γ}(l_{2})} ≤ ε_{0} = ε_{0}(d, p, n, γ, δ, K) ∀ t.

Next, observe that, for a_{m}(t, x) = a(t/m^{2}, x/m) and m ≥ 1, we have

||a_{m}(t, ·) − a_{m}(t, 0)||_{B^{|n|+γ}} ≤ ε + m^{−[(|n|+γ)∧1]}K

(K comes from Assumption 5.5), and for |n| + γ = 0 we can even drop the second term on the right. An analogous inequality holds for σ. It follows that, for ε sufficiently small and m sufficiently large, the statements of the lemma are true if we replace a, σ, w_{t}, f, g, and T in equation (5.1) by a_{m}, σ_{m}, mw_{t/m^{2}}, m^{−2}f(t/m^{2}, x/m), m^{−1}g(t/m^{2}, x/m), and m^{2}T, respectively. After this, it remains only to fix an appropriate m and make an obvious change of the unknown function in the above-mentioned modification of equation (5.1) (and use 1 − Δ ∼ m^{2} − Δ so that the norms || · ||_{n,p} of u(t/m^{2}, x/m) and u(t, x) are comparable). The lemma is proved.

Finally, we need the following result from [17], which, in a sense, is essentially covered by Theorem 2.4.7 from [34].

Lemma 6.7. Let δ > 0 and let ζ_{k} ∈ C^{∞}, k = 1, 2, 3, .... Assume that for any multi-index α and x ∈ ℝ^{d},

sup_{x∈ℝ^{d}} ∑_{k} |D^{α}ζ_{k}(x)| ≤ M(α),

where M(α) are some constants. Then there exists a constant N = N(d, n, M) such that, for any f ∈ H_{p}^{n},

∑_{k} ||ζ_{k}f||_{H_{p}^{n}}^{p} ≤ N||f||_{H_{p}^{n}}^{p}.

If in addition

∑_{k} |ζ_{k}(x)|^{p} ≥ δ,

then for any f ∈ H_{p}^{n},

||f||_{n,p}^{p} ≤ N(d, n, M, δ) ∑_{k} ||ζ_{k}f||_{n,p}^{p}.

Remark 6.8. We will also use a natural extension of this lemma to the case of Banach– space valued f.

Proof of Theorem 5.1. By Theorem 2.1, for any nonrandom z ∈ H_{p}^{n+2−2/p} there exists a unique solution u ∈ H_{p}^{n+2}(τ) of the equation du = Δu dt with initial condition z. This theorem also provides estimate (2.7) of the norm of u. From this theorem and the estimate, it follows that there exists a unique solution ū ∈ H_{p}^{n+2}(τ) of the equation du = Δu dt with initial condition u_{0} and

||ū||_{H_{p}^{n+2}(τ)}^{p} ≤ NE||u_{0}||_{n+2−2/p,p}^{p}.

After replacing the unknown function u with v + ū, it becomes obvious that, to solve (5.1) in H_{p}^{n+2}(τ) with the initial condition u_{0}, it suffices to solve the equation

\begin{matrix}du(t, x) = & [a^{ij}(t, x)u_{x^{i}x^{j}}(t, x) + f̄(u, t, x)] dt \\ + [σ^{ik}(t, x)u_{x^{i}}(t, x) + ḡ^{k}(u, t, x)] dw_{t}^{k} \end{matrix}

in the space H_{p,0}^{n+2}(τ), where (remember dū = Δū dt)

f̄(u, t, x) = {f(u + ū, t, x) + a^{ij}(t, x)ū_{x^{i}x^{j}}(t, x) − Δū}I_{A}(t),
ḡ^{k}(u, t, x) = {σ^{ik}(t, x)ū_{x^{i}}(t, x) + g^{k}(u + ū, t, x)}I_{A}(t),

and A is the set of ω, t for which ū(t, ·) ∈ H_{p}^{n+2}.

From Lemma 5.2, it follows that f̄ and ḡ satisfy the same conditions as f and g. This allows us to assume that u_{0} = 0. Furthermore, one can always extend f, g after time τ by equaling them to zero. That is why without loss of generality we assume that u_{0} = 0 and τ = T.

Now, define

Lu = a^{ij}(t, x)u_{x^{i}x^{j}}(t, x), Λu = σ^{i}(t, x)u_{x^{i}}(t, x),

and let {ζ_{k} : k = 1, 2, 3, ...} be a standard partition of unity such that, for any k, the support of ζ_{k} lies in a ball B_{k} of radius (1/4)κ_{ε/2}, where κ_{ε} is taken from Assumption 5.2 and ε from Lemma 6.6. Also for any k, we take a function η_{k} ∈ C_{0}^{∞} such that η_{k} = 1 on B_{k}, η_{k} = 0 outside doubled B_{k}, and 0 ≤ η_{k} ≤ 1. Denote by x_{k} the center of B_{k}, define L_{k}(t, x) = η_{k}(x)L(t, x) + (1 − η_{k}(x))L(t, x_{k}), and similarly define Λ_{k}.

Observe that, for any k, the operators L_{k}, Λ_{k} satisfy condition (i) of Lemma 6.6. Therefore, if we denote (f_{k}, g_{k}) := (L_{k}, Λ_{k})(uζ_{k}), then by this lemma

||uζ_{k}||_{H_{p}^{n+2}(T)} ≤ N||(f_{k}, g_{k})||_{F_{p}^{n}(T)} = N||(L_{k}, Λ_{k})(uζ_{k})||_{F_{p}^{n}(T)}.

Furthermore, by using that η_{k} = 1 everywhere where ζ_{k}u ≠ 0 we easily check that

(L_{k}, Λ_{k})(uζ_{k}) = (L, Λ)(uζ_{k}) = ζ_{k}(L, Λ)u + (uLζ_{k} + 2ζ_{kx} · au_{x}, uΛζ_{k}),

so that

||uζ_{k}||_{H_{p}^{n+2}(T)} ≤ N||ζ_{k}(L, Λ)u||_{F_{p}^{n}(T)} + N||(uLζ_{k} + 2ζ_{kx} · au_{x}, uΛζ_{k})||_{F_{p}^{n}(T)}.

We sum up the pth powers of the extreme terms and apply Lemma 6.7 and Lemma 5.2 in the estimates like the following one:

∑_{k} ||ua^{ij}ζ_{kx^{i}x^{j}}||_{n,p} ≤ N||ua^{ij}||_{n,p} ≤ N||u||_{n,p}.

Then we conclude that, for any u ∈ H_{p,0}^{n+2}(T),

||u||_{H_{p}^{n+2}(T)} ≤ N||(L, Λ)u||_{F_{p}^{n}(T)} + N||u||_{ℍ_{p}^{n+1}(T)}.  (6.4)

Next, we show that the last term on the right can be dropped. Indeed, ||u||_{n+1,p} ≤ ε||u_{xx}||_{n,p} + N(ε, d, p)||u||_{n,p}. Therefore, (6.4) can be modified by replacing the last term with N||u||_{ℍ_{p}^{n}(T)}. This can be done with any t ≤ T in place of T. After this, by Theorem 3.7, the inequality

E||u(t, ·)||_{n,p}^{p} ≤ N||(L, Λ)u||_{F_{p}^{n}(T)}^{p} + N ∫_{0}^{t} E||u(s, ·)||_{n,p}^{p} ds

holds for any t ≤ T. By Gronwall’s inequality, this yields that

||u||_{ℍ_{p}^{n}(T)}^{p} ≤ N||(L, Λ)u||_{F_{p}^{n}(T)}^{p}

which, along with the modified (6.4), proves that

||u||_{H_{p}^{n+2}(T)} ≤ N||(L, Λ)u||_{F_{p}^{n}(T)}  (6.5)

for any u ∈ H_{p,0}^{n+2}(T).

This is an a priori estimate. Now we use the standard method of continuity. For λ ∈ [0, 1] we consider the equation

du = (L_{λ}u + f) dt + (Λ_{λ}^{k}u + g^{k}) dw_{t}^{k}  (6.6)

with zero initial condition, where

L_{λ}u = λΔ + (1 − λ)L, Λ_{λ}u = (1 − λ)Λ

and (f, g) is an arbitrary element in F_{p}^{n}(T).

Observe that the a priori estimate (6.5) holds with the same constant N for all L_{λ}, Λ_{λ} in place of L, Λ. Next, take a λ_{0} ∈ [0, 1] and assume that for λ = λ_{0} equation (6.6) with zero initial data has a unique solution u ∈ H_{p,0}^{n+2}(T) for any (f, g) ∈ F_{p}^{n}(T). By the way, this assumption is satisfied for λ_{0} = 1 by Theorem 6.3. Then we have an operator

R_{λ_{0}} : F_{p}^{n}(T) → H_{p,0}^{n+2}(T)

such that R_{λ_{0}}(f, g) = u. From (6.5) we get that

||R_{λ_{0}}(f, g)||_{H_{p}^{n+2}(T)} ≤ N||(f, g)||_{F_{p}^{n}(T)}.  (6.7)

For other λ ∈ [0, 1] we rewrite (6.6) as

du = (L_{λ_{0}}u + {(λ − λ_{0})(Δ − L)u + f}) dt + (Λ_{λ_{0}}^{k}u + {(λ_{0} − λ)Λ^{k}u + g^{k}}) dw_{t}^{k}

and we solve the last equation by iterations. Define u_{0} = 0 and

u_{j+1} = R_{λ_{0}}((λ − λ_{0})(Δ − L)u_{j} + f, (λ_{0} − λ)Λu_{j} + g).

Then by (6.7)

||u_{j+1} − u_{j}||_{H_{p}^{n+2}(T)} ≤ N|λ − λ_{0}| ||((Δ − L)(u_{j} − u_{j−1}), Λ(u_{j} − u_{j−1}))||_{F_{p}^{n}(T)}
≤ N_{1}|λ − λ_{0}| ||u_{j} − u_{j−1}||_{H_{p}^{n+2}(T)},

where N_{1} is independent of j, λ, and λ_{0}. If N_{1}|λ − λ_{0}| ≤ 1/2, then u_{j} is a Cauchy sequence in H_{p}^{n+2}(T), which converges by Theorem 3.7. Its limit satisfies

u = R_{λ_{0}}((λ − λ_{0})(Δ − L)u + f, (λ_{0} − λ)Λu + g),

which is equivalent to (6.6).

In this way we show that if (6.6) is solvable for a λ_{0}, then it is solvable for λ satisfying N_{1}|λ − λ_{0}| ≤ 1/2. In finite number of steps starting with λ = 1, we get to λ = 0. This proves the theorem if f, g are independent of u.

To consider general f and g, it remains only to repeat the proof of Theorem 6.4 taking f(u, t, x) and g(u, t, x) instead of f + L_{1}(u) and g + Λ_{1}(u) there. The theorem is proved.

7. Embedding Theorems for H_{p}^{n}(τ)

The above theory of solvability looks satisfactory only until one tries to apply it to concrete problems when one is interested in getting not only solvability but also some qualitative properties of solutions, like continuity, decay at infinity, compactness of support, and so on. To answer such questions, one has to understand what qualitative properties the solutions have. Since solutions are just arbitrary functions from H_{p}^{n}(τ), we are actually interested in properties of functions from this space. Let us fix a T ∈ [0, ∞) and a stopping time τ ≤ T.

The first two assertions of the following theorem are straightforward corollaries of two Sobolev’s theorems. One says that H_{p}^{n} ⊂ C^{α} if α := n − d/p > 0, where C^{α} = C^{α}(ℝ^{d}) is the Zygmund space (which differs from the usual Hölder space C^{α} = C^{α}(ℝ^{d}) only if α is an integer, see [33]). The second one says that H_{p}^{n} ⊂ H_{q}^{m} if m < n and n − d/p = m − d/q.

Theorem 7.1. (i) If α := n − d/p > 0 and u ∈ H_{p}^{n}(τ), then u ∈ L_{p}(^{|}(0, τ]], C^{α}), where C^{α} is the Zygmund space. In addition,

E ∫_{0}^{τ} ||u(t, ·)||_{C^{α}}^{p} dt ≤ N(d, n, p)||u||_{H_{p}^{n}(τ)}^{p}.
(ii) If m < n, n − d/p = m − d/q, and u ∈ H_{p}^{n}(τ), then
E ∫_{0}^{τ} ||u(t, ·)||_{m,q}^{p} dt ≤ N(d, n, m, p)||u||_{H_{p}^{n}(τ)}^{p}.

7. EMBEDDING THEOREMS FOR H_{p}^{n}(τ) 219

(iii) For any function u ∈ H_{2}^{n}(τ), we have u ∈ C([0, τ], H_{2}^{n−1}) (a.s.) and

E sup_{t≤τ} ||u(t, ·)||_{n−1,2}^{2} ≤ N(d, n, T)||u||_{H_{2}^{n}(τ)}^{2}.

Proof. As we have said before the theorem, we only need to prove the third assertion. By Remark 3.4, we may assume that n = 1. Denote u_{0} = u(0), v = T_{t}u_{0}. Observe that by Theorem 2.1 we have v ∈ H_{2}^{1}(τ). By Minkowski’s inequality, ||ζ ∗ u||_{p} ≤ ||u||_{p}||ζ||_{1}, so that

||T_{t}u_{0}||_{2} ≤ ||u_{0}||_{2}, E sup_{t} ||v(t, ·)||_{2}^{2} ≤ E||u_{0}||_{2}^{2}.

In addition, almost obviously, T_{t}u_{0} is a continuous (analytic) L_{2}–valued function in t for t > 0. Also, we have (T_{t}u_{0})′_{t} = T_{t}Δu_{0}, which implies that ||T_{t}u_{0} − u_{0}||_{2} ≤ t||Δu_{0}||_{2} → 0 if u_{0} ∈ H_{2}^{2}. Adding that the set H_{2}^{2} is dense in L_{2}, we conclude that T_{t} is a continuous semigroup in L_{2}. This means that v ∈ C([0, τ], H_{2}^{n−1}) and shows that we need only to consider u − v. In other words, in the rest of the proof we may and will assume that u_{0} = 0.

In this case, denote f = (𝔻u − Δu)I_{t≤τ}, and g = (𝕊u)I_{t≤τ}. Solve equation (4.2) on [0, ∞) with zero initial data. By uniqueness, the solution coincides with u on [0, τ]. By Theorem 4.2, assertion (iii) holds. The theorem is proved.

Further results about some basic properties of the spaces H_{p}^{n}(τ) are collected in the following theorem. As we have seen in Theorem 4.2, assertion (iii) of Theorem 7.1 is true not only for p = 2 but also for any p ≥ 2. However, for p > 2 a much stronger statement (i) of Theorem 7.2 holds.

Theorem 7.2. (i) If p > 2, 1/2 > β > α > 1/p, then for any function u ∈ H_{p}^{n}(τ), we have u ∈ C^{α−1/p}([0, τ], H_{p}^{n−2β}) (a.s.) and for any stopping time η ≤ τ,

E||u(t ∧ η, ·) − u(s ∧ η, ·)||_{n−2β,p}^{p} ≤ N(d, β, p, T)|t − s|^{βp−1}||u||_{H_{p}^{n}(τ)}^{p} ∀t, s ≤ T; (7.1)

E||u(t, ·)||_{C^{α−1/p}([0,τ],H_{p}^{n−2β})}^{p} ≤ N(d, β, α, p, T)||u||_{H_{p}^{n}(τ)}^{p}.  (7.2)

(ii) If q ≥ p > 2 and θ ∈ (0, 1), then for

m < n + \frac{d}{q} − \frac{d + 2(1 − θ)}{p}, u ∈ H_{p}^{n}(τ),

we have u ∈ L_{p/θ}((0, τ), H_{q}^{m}) (a.s.) and

E(∫_{0}^{τ} ||u(t, ·)||_{m,q}^{p/θ} dt)^{θ} ≤ N(d, p, q, n, m, θ, T)||u||_{H_{p}^{n}(τ)}^{p}.

In particular (take θ = p/q),

E(∫_{0}^{τ} ||u(t, ·)||_{m,q}^{q} dt)^{p/q} ≤ N(d, p, q, n, m, T)||u||_{H_{p}^{n}(τ)}^{p}

if

q > p > 2, m < n − (d + 2)(\frac{1}{p} − \frac{1}{q}).

To prove the theorem, we need two lemmas. Remember the notation T_{t} introduced in (2.3).

Lemma 7.3. For any h ∈ L_{p}, θ ∈ [0, 1], and t > 0, we have
||e^{−t}T_{t}h||_{p} ≤ N \frac{1}{t^{θ}} ||h||_{−2θ,p}, ||(T_{t} − 1)h||_{p} ≤ Nt^{θ}||h||_{2θ,p},  (7.3)

where N = N(d, p, θ).

Proof. These inequalities follow from Theorem 14.11 of [14]. For the sake of completeness, we prove them. The derivative (T_{t}h)′_{t} can be represented as t^{−1} times a convolution of h with a function having finite L_{1}–norm, the norm being independent of t. By Minkowski’s inequality, ||ζ ∗ h||_{p} ≤ ||h||_{p}||ζ||_{1} so that

||(T_{t}h)′_{t}||_{p} ≤ N(d, p)\frac{1}{t} ||h||_{p}, ||(e^{−t}T_{t}h)′_{t}||_{p} ≤ N(d, p)\frac{1}{t} ||h||_{p}.  (7.4)

Hence for θ ∈ (0, 1), (see (2.2))

||(1 − Δ)^{θ}(e^{−t}T_{t}h)||_{p} = c(θ)|| ∫_{0}^{∞} \frac{e^{−(s+t)}T_{s+t}u − e^{−t}T_{t}h}{s^{θ}} \frac{ds}{s}||_{p}

≤ N ∫_{0}^{t} \frac{1}{s^{θ}} ||e^{−(s+t)}T_{s+t}u − e^{−t}T_{t}h||_{p} \frac{ds}{s} + N ∫_{t}^{∞} \frac{ds}{s^{1+θ}} ||h||_{p}.

Here by (7.4)

||e^{−(s+t)}T_{s+t}u − e^{−t}T_{t}h||_{p} ≤ N \frac{s}{t} ||h||_{p},

∫_{0}^{t} \frac{1}{s^{θ}} ||e^{−(s+t)}T_{s+t}u − e^{−t}T_{t}h||_{p} \frac{ds}{s} ≤ ||h||_{p} ∫_{0}^{t} \frac{1}{ts^{θ}} ds = N||h||_{p} \frac{1}{t^{θ}} .

This proves that

||e^{−t}T_{t}(1 − Δ)^{θ}h||_{p} ≤ N||h||_{p}t^{−θ},  (7.5)

which gives the first inequality in (7.3) after replacing (1 − Δ)^{θ}h with h. If θ = 0, then one can take N = 1 in the first inequality in (7.3), which follows from Minkowski’s inequality. If θ = 1, then we use (7.4) and the fact that (e^{−t}T_{t}h)′_{t} = e^{−t}T_{t}(Δ − 1)h.

To prove the second inequality in (7.3) for t ∈ [0, 1] and θ > 0, it suffices to notice that (T_{t}h)′_{t} = ΔT_{t}h and Δ = [Δ(1 − Δ)^{−1}](1 − Δ)^{1−θ}(1 − Δ)^{θ} and use (7.5) in the following estimates

||(T_{t} − 1)h||_{p} ≤ ∫_{0}^{t} ||[Δ(1 − Δ)^{−1}](1 − Δ)^{1−θ}T_{s}[(1 − Δ)^{θ}h]||_{p} ds

≤ N ∫_{0}^{t} ||(1 − Δ)^{1−θ}T_{s}[(1 − Δ)^{θ}h]||_{p} ds ≤ N ∫_{0}^{t} s^{θ−1} ds||h||_{2θ,p} = Nt^{θ}||h||_{2θ,p}.

For t ≥ 1 or θ = 0, the second inequality in (7.3) is trivial since ||h||_{p} ≤ ||h||_{2θ} and t^{0} = 1. The lemma is proved.

Lemma 7.4. Let αp > 1 and p ≥ 1. Then, for any continuous L_{p}-valued function h(t), and s ≤ t, we have

\begin{matrix}||h(t) − h(s)||_{p}^{p} ≤ N(α, p)(t − s)^{αp−1} ∫_{s}^{t} ∫_{s}^{t} I_{r_{2}>r_{1}} \frac{||h(r_{2}) − h(r_{1})||_{p}^{p}}{|r_{2} − r_{1}|^{1+αp}} dr_{1}dr_{2} \\ = N(α, p)(t − s)^{αp−1} ∫_{0}^{t−s} \frac{dγ}{γ^{1+αp}} ∫_{s}^{t−γ} ||h(r + γ) − h(r)||_{p}^{p} dr (\frac{0}{0} := 0). \end{matrix}  (7.6)

This is one of embedding theorems for Slobodetskii’s spaces (see, for instance, [33]). It is to be said that embedding theorems were applied for studying continuity properties of random processes for quite long time; see, for instance, [1], [9], [11], [19], [35]. The following consequence of (7.6),

E sup_{0≤s<t≤T} \frac{||h(t) − h(s)||_{p}^{p}}{(t − s)^{αp−1}} ≤ N(α, p) ∫_{0}^{T} ∫_{0}^{T} I_{r_{2}>r_{1}} \frac{E||h(r_{2}) − h(r_{1})||_{p}^{p}}{|r_{2} − r_{1}|^{1+αp}} dr_{1}dr_{2}, (7.7)

can be used wherever one uses Kolmogorov’s continuity criterion (see, [1]). By the way, embedding theorems are (for the most part) known for multi–dimensional case as well, and one can use them for studying random fields (cf. [35]).

In [19], equation (7.7) with T = π was used to prove that Fourier’s series

\frac{1}{\sqrt{π}} tη_{0} + \sqrt{\frac{2}{π}} ∑_{n=1}^{∞} η_{n} \frac{1}{n} sin nt,

where η_{n} are N(0, 1) i.i.d. random variables, converges uniformly on [0, π]. This is the way the Wiener process is introduced in [19]. Estimate (7.7) also gives a very good modulus of continuity for the Wiener process. Of course, an advantage of normal centered variables is that (E|ξ|^{p})^{1/p} = N(p)(E|ξ|^{2})^{1/2}. Instead of this, in our applications here we use Burkholder–Davis–Gundy inequalities. By the way, a very short proof of (7.6) may be found in [19]. Finally, notice that the space L_{p} in (7.6) and (7.7) can be replaced with any Banach space. Proof of Theorem 7.2. Take u ∈ H_{p}^{n}(τ), and define f = (𝔻u−Δu)I_{t≤τ} and g = (𝕊u)I_{t≤τ}. Notice that the function u on [0, τ] satisfies the equation

dv = (Δv + f) dt + g^{k} dw_{t}^{k}.  (7.8)

By Theorem 5.1, equation (7.8) on [0, T] with initial condition v(0) = u(0) has a unique solution v ∈ H_{p}^{n}(T). The difference u−v satisfies the heat equation on [0, τ] with zero initial condition. It follows that u(t, ·) = v(t, ·) on [0, τ], and by Theorem 5.1,

||v||_{H_{p}^{n}(T)} ≤ N(||f||_{ℍ_{p}^{n−2}(τ)} + ||g||_{ℍ_{p}^{n−1}(τ,l_{2})} + (E||u(0)||_{n−2/p,p}^{p})^{1/p}) ≤ N||u||_{H_{p}^{n}(τ)}.

The inequality ||v||_{H_{p}^{n}(T)} ≤ N||u||_{H_{p}^{n}(τ)} and the equality u(t, ·) = v(t, ·) on [0, τ] show that we only need to prove the theorem for v, T in place of u, τ. In particular, we may and will assume that τ = T. Also, observe that

||f||_{ℍ_{p}^{n−2}(T)} + ||g||_{ℍ_{p}^{n−1}(T,l_{2})} + (E||u(0)||_{n−2/p,p}^{p})^{1/p} ≤ N||u||_{H_{p}^{n}(T)}.

This allows us to concentrate only on solutions of equations (7.8) on [0, T] and to prove our assertions with ||(f, g)||_{F_{p}^{n−2}(T)} + (E||u(0)||_{n−2/p,p}^{p})^{1/p} in place of ||u||_{H_{p}^{n}(T)}. Therefore, below we take τ = T and take the function u ∈ H_{p}^{n}(T) as a solution of (7.8). As in the proof of Theorem 4.2, we may and will assume that f and g are as in (4.7) and u(0) ∈ C_{0}^{∞}.

After these preparations, we are going to prove assertion (i). By Remark 3.4, without loss of generality we assume that n = 2β. Denote

u_{1}(t) = T_{t}u_{0} + ∫_{0}^{t} T_{t−s}f(s) ds.

It is easy to see that

u_{1}(r + γ) − u_{1}(r) = (T_{γ} − 1)u_{1}(r) + ∫_{0}^{γ} T_{γ−ρ}f(r + ρ) dρ.

Therefore, E||u_{1}(r + γ) − u_{1}(r)||_{p}^{p} ≤ N(A_{1}(r, γ) + B_{1}(r, γ)), where

A_{1}(r, γ) = E||(T_{γ} − 1)u_{1}(r)||_{p}^{p}, B_{1}(r, y) = E|| ∫_{0}^{γ} T_{γ−ρ}f(r + ρ) dρ||_{p}^{p},

and, by Lemma 7.4,

E||u_{1}(t) − u_{1}(s)||_{p}^{p} ≤ N(t − s)^{αp−1}(I_{1}(t, s) + J_{1}(t, s)),

222 5. ANALYTIC APPROACH TO SPDE’S

with

I_{1}(t, s) = ∫_{0}^{t−s} \frac{dγ}{γ^{1+αp}} ∫_{s}^{t−γ} A_{1}(r, γ) dr, J_{1}(t, s) = ∫_{0}^{t−s} \frac{dγ}{γ^{1+αp}} ∫_{s}^{t−γ} B_{1}(r, γ) dr.

By using Hölder’s inequality and Lemma 7.3 and observing that (β − 1)q > −1 for q = p/(p − 1), we get (remember n = 2β)

B_{1}(r, γ) = E ∫_{ℝ^{d}} | ∫_{0}^{γ} ρ^{β−1}ρ^{1−β}T_{ρ}f(r + γ − ρ) dρ|^{p} dx

≤ (∫_{0}^{γ} ρ^{(β−1)q} dρ)^{p/q}E ∫_{0}^{γ} ρ^{(1−β)p} ∫_{ℝ^{d}} |T_{ρ}f(r + γ − ρ)|^{p} dxdρ
≤ Nγ^{βp−1}E ∫_{0}^{γ} ||f(r + γ − ρ)||_{n−2,p}^{p} dρ = Nγ^{βp−1}E ∫_{0}^{γ} ||f(r + ρ)||_{n−2,p}^{p} dρ.

This and the inequality α < β implies that

J_{1}(t, s) ≤ N ∫_{0}^{t−s} \frac{dγ}{γ^{2+(α−β)p}} ∫_{0}^{γ} dρE ∫_{s}^{t−γ} ||f(r + ρ)||_{n−2,p}^{p} dr

≤ N ∫_{0}^{t−s} \frac{dγ}{γ^{2+(α−β)p}} ∫_{0}^{γ} dρE ∫_{0}^{t} ||f(r)||_{n−2,p}^{p} dr

= N(t − s)^{(β−α)p}E ∫_{0}^{t} ||f(r)||_{n−2,p}^{p} dr.

To estimate I_{1}, we use Lemma 7.3 and Theorem 2.1 to obtain A_{1}(r, γ) ≤ Nγ^{βp}E||u_{1}(r)||_{n,p}^{p} ≤ Nγ^{βp}{E||u(0)||_{n−2/p,p}^{p} + E ∫_{0}^{r} ||f(s)||_{n−2,p}^{p} ds}

I_{1}(t, s) ≤ ∫_{0}^{t−s} \frac{dγ}{γ^{1+αp}} ∫_{0}^{t} A_{1}(r, γ) dr

≤ N(t − s)^{(β−α)p}{E||u(0)||_{n−2/p,p}^{p} + E ∫_{0}^{t} ||f(r)||_{n−2,p}^{p} dr}.

For u_{2} := u − u_{1} we have

u_{2}(r + γ) − u_{2}(r) = (T_{γ} − 1)u_{2}(r) + ∫_{r}^{r+γ} T_{r+γ−ρ}g^{k}(ρ) dw_{ρ}^{k},
E||u_{2}(r + γ) − u_{2}(r)||_{p}^{p} ≤ N(A_{2}(r, γ) + B_{2}(r, γ)),

where

A_{2}(r, γ) = E||(T_{γ} − 1)u_{2}(r)||_{p}^{p}, B_{2}(r, γ) = E|| ∫_{r}^{r+γ} T_{r+γ−ρ}g^{k}(ρ) dw_{ρ}^{k}||_{p}^{p}.

By Lemma 7.4,

E||u_{2}(t) − u_{2}(s)||_{p}^{p} ≤ N(t − s)^{αp−1}(I_{2}(t, s) + J_{2}(t, s)),

with

I_{2}(t, s) = ∫_{0}^{t−s} \frac{dγ}{γ^{1+αp}} ∫_{s}^{t−y} A_{2}(r, γ) dr, J_{2}(t, s) = ∫_{0}^{t−s} \frac{dγ}{γ^{1+αp}} ∫_{s}^{t−γ} B_{2}(r, γ) dr.

Here we use Lemma 7.3, the Burkholder–Davis–Gundy inequalities, and the inequality (2β − 1)q > −1, where q = p/(p − 2). Then, similarly to the above calculations, we obtain

B_{2}(r, γ) ≤ NE ∫_{ℝ^{d}}[∫_{0}^{γ} ρ^{2β−1}ρ^{1−2β}||T_{ρ}g(r + γ − ρ)||_{l_{2}}^{2} dρ]^{p/2} dx

≤ Nγ^{βp−1}E ∫_{0}^{γ} ρ^{(1−2β)p/2}||T_{ρ}g(r + γ − ρ)||_{p}^{p} dρ

≤ Nγ^{βp−1}E ∫_{0}^{γ} ||(1 − Δ)^{β−1/2}g(r + γ − ρ)||_{p}^{p} dρ,

J_{2}(t, s) ≤ NE ∫_{0}^{t−s} \frac{dγ}{γ^{2+(α−β)p}} ∫_{s}^{t−γ} dr ∫_{0}^{γ} ||g(r + γ − ρ)||_{n−1,p}^{p} dρ

≤ NE ∫_{0}^{t−s} \frac{dγ}{γ^{2+(α−β)p}} ∫_{0}^{γ} dρ ∫_{0}^{t} ||g(r)||_{n−1,p}^{p} dr

= N(t − s)^{(β−α)p}E ∫_{0}^{t} ||g(r)||_{n−1,p}^{p} dr.

Finally, again by Lemma 7.3 and Theorem 4.2, we conclude

I_{2}(t, s) ≤ NE ∫_{0}^{t−s} \frac{dγ}{γ^{1+(α−β)p}} ∫_{s}^{t−γ} ||u_{2}(r)||_{2β,p}^{p} dr

≤ N(t − s)^{(β−α)p}E ∫_{0}^{t} ||u_{2}(r)||_{n,p}^{p} dr ≤ N(t − s)^{(β−α)p}E ∫_{0}^{t} ||g(r)||_{n−1,p}^{p} dr.

Collecting all these estimates, we get (7.1) at least for η = T (= τ). In the general case, observe that if u(t ∧ η) − u(s ∧ η) is not zero, then s ∧ η = s and s ≤ s ∧ η ≤ t ∧ η ≤ t. After this it suffices to notice that, instead of points t and s on the left in (7.6), we can obviously take any two points between them.

The proof of (7.2) goes exactly the same way, the only difference being that this time we use (7.7).

To prove assertion (ii) notice that from an interpolation theorem (Theorem 2.4.2 in [33]) we have

||u||_{m(θ)−d/p+d/q,q} ≤ N(d, p, q, m(0), m(1), θ)||u||_{m(0),p}^{1−θ}||u||_{m(1),p}^{θ}

whenever 1 < p < q < ∞, θ ∈ (0, 1), m(θ) := (1 − θ)m(0) + θm(1) ≠ m(0), m(i) ≤ n, and u ∈ H_{p}^{n}. Theorem 14.2 from [14] shows that the case p = q actually need not be excluded. Note also that, under the conditions in (ii), there is a β such that 1/2 > β > 1/p and m ≤ n − 2β(1 − θ) − d/p + d/q = m(θ) − d/p + d/q, where m(θ) := (1 − θ)(n − 2β) + θn. Therefore,

E(∫_{0}^{T} ||u(t, ·)||_{m,q}^{p/θ} dt)^{θ} ≤ E(∫_{0}^{T} ||u(t, ·)||_{m(θ)−d/p+d/q,q}^{p/θ} dt)^{θ}

≤ NE(∫_{0}^{T} ||u(t, ·)||_{n−2β,p}^{(1−θ)p/θ}||u(t, ·)||_{n,p}^{p} dt)^{θ}

≤ NE sup_{t≤T} ||u(t, ·)||_{n−2β,p}^{(1−θ)p}(∫_{0}^{T} ||u(t, ·)||_{n,p}^{p} dt)^{θ}.

To prove (ii), it remains only to apply Hölder’s inequality, (3.4), and (7.2) with an α such that β > α > 1/p. The theorem is proved.

8. Applications

8.1. Filtering Equation. Take p ≥ 2 and an integer d_{1} > d. Let (Ω, F, P) be a complete probability space and let w_{t} be a d_{1}–dimensional Wiener process on this space. Consider a d_{1}–dimensional two–component process z_{t} = (x_{t}, y_{t}) with x_{t} being d–dimensional and y_{t} d_{1} − d–dimensional. We assume that z_{t} is defined as a solution of the system

\begin{matrix}dx_{t} = & b(t, z_{t}) dt + θ(t, z_{t}) dw_{t}, \\ dy_{t} = & B(t, z_{t}) dt + Θ(t, y_{t}) dw_{t} \end{matrix}  (8.1)

with some initial data. The coefficients of (8.1) are assumed to be vectors or matrices of appropriate dimensions satisfying the following assumptions.

Assumption 8.1. The functions b, θ, B, Θ are Borel measurable and bounded functions of their arguments. Each of them satisfies the Lipschitz condition with respect to z with constant K. Moreover, θ(t, x, y) is continuously differentiable with respect to x (not z) and its first derivatives with respect to x^{i}s are continuous in y and satisfy the Lipschitz condition, with constant K, with respect to x (not z).

Assumption 8.2. The symmetric matrix ΘΘ^{∗} is invertible and Ψ := (ΘΘ^{∗})^{−1/2} is a bounded function (of (t, y)).

Assumption 8.3. For any ξ ∈ ℝ^{d} and z = (x, y) ∈ ℝ^{d_{1}} and t > 0,
|Q(t, y)θ^{∗}(t, z)ξ|^{2} ≥ δ|ξ|^{2},

where Q is the orthogonal projector on ker Θ. In other words

(θ(1 − Θ^{∗}Ψ^{2}Θ)θ^{∗}ξ, ξ) ≥ δ|ξ|^{2}.

One can easily check that this assumption is satisfied if the diffusion matrix of system (8.1) is block upper–triangular:

( \begin{matrix}θ \\ Θ \end{matrix} ) = ( \begin{matrix}θ_{1} & θ_{2} \\ 0 & Θ_{2} \end{matrix} )

and the d × d matrix θ_{1} is non-degenerate so that |θ_{1}^{∗}ξ|^{2} ≥ δ|ξ|^{2}.

We also make the following assumption regarding the initial data for (8.1).

Assumption 8.4. The random vectors x_{0}, y_{0} are independent of the process w_{t}. The conditional distribution of x_{0} given y_{0} has a density, which we denote by π_{0}(x) = π_{0}(ω, x). We have π_{0} ∈ L_{p}(Ω, H_{p}^{2−2/p}).

Let

a(t, x) = (1/2)θθ^{∗}(t, x, y_{t}), σ(t, x) = θΘ^{∗}Ψ(t, x, y_{t}),
β(t, x) = ΨB(t, x, y_{t}), β_{t} = β(t, x_{t}),

and let β^{k} be the kth coordinate of the vector β where k = 1, ..., d_{1} − d. Also, for a function u define

Lu = ∑_{i,j≤d}(a^{ij}u)_{x^{i}x^{j}} − ∑_{i≤d}(b^{i}u)_{x^{i}}, Λ^{k}u = β^{k}u − ∑_{i≤d}(σ^{ik}u)_{x^{i}}.

Finally, let F_{t}^{y} = σ{y_{s} : s ≤ t}.

Theorem 8.1. Under the above assumptions there exists π ∈ ∩_{T >0}H_{p}^{2}(T) such that for any t ≥ 0, the conditional distribution of x_{t} given F_{t}^{y} has a density coinciding with π(t, ·) almost surely. In addition, π satisfies the equation

dπ = Lπ dt + ∑_{k≤d_{1}−d}[Λ^{k}π − β̄_{t}^{k}π] {Ψ^{k}Θ dw_{t} + [β_{t}^{k} − β̄_{t}^{k}] dt},  (8.2)

with initial data π(0, ·) = π_{0}, where the missing arguments are t, x, y_{t}, vector Ψ^{k} is the kth row of Ψ, and β̄_{t} := (π(t, ·), β(t, ·, y_{t})).

To prove the theorem, we need a lemma, but first we make some comments. Notice that a little bit strange way of writing equation (8.2), with dt in two terms, is related to the fact that

w̄_{t} := ∫_{0}^{t}{ΨΘ dw_{s} + [β_{s} − β̄_{s}] ds}

is a d_{1}−d–dimensional F_{t}^{y}–adapted Wiener process (the so–called innovation process). Here we do not need this fact.

We will only use that in (8.2) we have Ψ^{k}Θ dw_{t} = dw̃_{t}^{k}, where w̃_{t} is a d_{1} −d–dimensional F_{t}–adapted Wiener process. This is easily proved by using Levy’s theorem, since it follows from the definition of Ψ that ⟨ΨΘ dw_{t}⟩ = Idt, where I is the d_{1} − d unit matrix. In turn, this shows that in notation (5.2) applied to (8.2) we have

α^{ij} = \frac{1}{2} ∑_{k≤d_{1}−d} σ^{ik}σ^{jk},

which, by Assumption 8.3, guarantees that Assumption 5.1 is satisfied for equation (8.3) below with a K and δ/2 instead of δ. Assumption 8.1 provides certain smoothness of the coefficients of (8.3), which leads us to the conclusion that the function π̄ considered in the following lemma exists by Theorem 5.1 and Remark 5.6. By the way, the requirement in Assumption 8.1 that first derivatives of θ(t, x, y) with respect to x^{i}s be continuous in y, which is not needed at this point, will be needed later in some passages to the limit. It is worth noting that this requirement follows automatically from other requirements in Assumption 8.1. Indeed, one knows that if a function f(x, y) is Lipschitz continuous in (x, y) and has bounded generalized second–order derivatives in x, then f_{x} is Hölder continuous in y with exponent 1/2.

Lemma 8.2. Let π̄ be a unique solution in ∩_{T} H_{p}^{1}(T) of the equation

dπ̄ = Lπ̄ dt + ∑_{k≤d_{1}−d} Λ^{k}π̄ {Ψ^{k}Θ dw_{t} + β_{t}^{k} dt}  (8.3)

with initial data π̄(0, ·) = π_{0}, where we use the same notation as in (8.2). Then π̄(t, ·) is F_{t}^{y}–measurable.

Proof. Observe that now we write dt in two terms on the right for the reason that Ψ^{k}Θ dw_{t} + β_{t}^{k} dt = Ψ^{k} dy_{t}, which makes the assertion quite natural since everything in (8.3) depends only on y_{t}. Further (cf. [24] or [30]), there is an F_{t}^{y}–adapted measurable process β̃_{t} = (β̃_{t}^{1}, ..., β̃_{t}^{d_{1}−d}) such that the process

w̃_{t} := ∫_{0}^{t}{ΨΘ dw_{s} + [β_{s} − β̃_{s}] ds}  (8.4)

is a d_{1}−d–dimensional F_{t}^{y}–adapted Wiener process (actually β̃_{t} is a modification of E{β_{t}|F_{t}^{y}}). By Itô’s formula the function π̃ defined by

ρ̃_{t}π̃(t, x) := π̄(t, x), ρ̃_{t} := exp\{∫_{0}^{t} β̃_{s} dw̃_{s} + \frac{1}{2} ∫_{0}^{t} |β̃_{s}|^{2} ds\}

satisfies

dπ̃ = Lπ̃ dt + ∑_{k≤d_{1}−d}[Λ^{k}π̃ − β̃_{t}^{k}π̃] dw̃_{t}^{k}  (8.5)

with initial data π̄(0, ·) = π_{0}. One can consider this equation on the probability space (Ω, F, P) with filtration F_{t}^{y} and Wiener process w̃_{t}. Let us denote by H̃_{p}^{n}(τ) the H_{p}^{n}(τ) spaces constructed on the basis of F_{t}^{y} and w̃_{t}. Define

τ_{n} := inf\{t ≥ 0 : ρ̃_{t} ≤ \frac{1}{n}\} ∧ n.

Observe that τ_{n} are F_{t}^{y}–stopping times. By the theory from previous sections, equation (8.5) with initial data u_{0} has a unique solution in H̃_{p}^{1}(τ_{n}) (even in H̃_{p}^{2}(τ_{n})). On the other hand, we can replace w̃_{t} in (8.5) with its value from (8.4) and get an equation in terms of the original H_{p}^{n}(τ_{n}). This equation has a unique solution in the original H_{p}^{1}(τ_{n}). Since H̃_{p}^{1}(τ_{n}) ⊂ H_{p}^{1}(τ_{n}), this implies that any H_{p}^{1}(τ_{n})–solution belongs to H̃_{p}^{1}(τ_{n}), and since π̃ is obviously a H_{p}^{1}(τ_{n})–solution, we get that π̃(t ∧ τ_{n}, ·) is F_{t}^{y}–adapted, which, for n → ∞, implies that π̃(t, ·) is F_{t}^{y}–adapted. Finally, ρ̃_{t} is F_{t}^{y}–adapted, and therefore π̄(t, ·) is F_{t}^{y}– adapted. The lemma is proved. Proof of Theorem 8.1. We imitate some proofs from [24] or [30] omitting some simple arguments which the reader can find there. First, we know (cf. Theorems 2.1 and 2.2 of [24]) that there is an F_{t}^{y}–adapted measure–valued process π_{t} such that, for any ϕ ∈ C_{0}^{∞} and t ≥ 0, we have E{ϕ(x_{t})|F_{t}^{y}} = (π_{t}, ϕ) (a.s.). We also know that π_{t} satisfies the following equation for any ϕ ∈ C_{0}^{∞}

d(π(t, ·), ϕ) = (π(t, ·), L^{∗}ϕ) dt + (π(t, ·), Λ^{k∗}ϕ − β̄_{t}^{k}ϕ) {Ψ^{k}Θ dw_{t} + [β_{t}^{k} − β̄_{t}^{k}] dt}, (8.6)

where β̄_{t} = (π(t, ·), β(t, ·, y_{t})) and L^{∗}, Λ^{∗} are the operators formally adjoint to L, Λ.

Now, let us assume for a moment that, in addition to our hypotheses, each derivative of b, B, θ with respect to x exists and is bounded. Notice that, by Sobolev’s embedding theorem, any measure on ℝ^{d} belongs to H_{p}^{−n} with n > (p − 1)d/p. This implies that π_{t} is a ∩_{T >0}H_{p}^{−n}(T)–solution of (8.6). If we look at β̄ as a given process, then by Theorem 5.1 and Remark 5.6 there is a unique solution of (8.6) in the class ∩_{T >0}H_{p}^{−n}(T) with the initial condition π(0, ·) = π_{0}. On the other hand, since the coefficients are infinitely differentiable (but π_{0} belongs only to L_{p}(Ω, H_{p}^{2−2/p})), by the same Theorem 5.1 and Remark 5.6 equation (8.6) has a solution in ∩_{T >0}H_{p}^{2}(T). Owing to the uniqueness, this solution is π. Thus π ∈ ∩_{T >0}H_{p}^{2}(T), which allows us to rewrite (8.6) as (8.2) after integrating by parts, and finishes the proof in the particular case of very smooth coefficients.

Passing to the general case notice that we only need to prove that π ∈ ∩_{T >0}H_{p}^{2}(T), since equation (8.2) is derived from (8.6) as above. Further, take a nonnegative function ζ ∈ C_{0}^{∞} with unit integral and for ε > 0 define ζ_{ε}(x) = ε^{−d}ζ(x/ε), h^{(ε)}(t, x, y) = h(t, x, y) ∗ ζ_{ε}(x), where the convolution is taken with respect to x. Consider the process z_{t}^{ε} = (x_{t}^{ε}, y_{t}^{ε}) defined as a solution of

\begin{matrix}dx_{t}^{ε} = & b^{(ε)}(t, z_{t}^{ε}) dt + θ^{(ε)}(t, z_{t}^{ε}) dw_{t}, \\ dy_{t}^{ε} = & B^{(ε)}(t, z_{t}^{ε}) dt + Θ(t, y_{t}^{ε}) dw_{t} \end{matrix}  (8.7)

with initial condition z_{0}^{ε} = z_{0}. If we denote F_{t}^{ε} = σ{y_{s}^{ε} : s ≤ t} and π^{ε}(t, x) the conditional distribution of x_{t}^{ε} given F_{t}^{ε}, then, by the above, π^{ε} ∈ ∩_{T >0}H_{p}^{2}(T) and π^{ε} satisfies the following version of (8.2)

dπ^{ε} = L_{ε}π^{ε} dt + ∑_{k≤d_{1}−d}[Λ_{εk}π^{ε} − β̄_{εt}^{k}π^{ε}] {Ψ_{ε}^{k}Θ_{ε} dw_{t} + [β_{εt}^{k} − β̄_{εt}^{k}] dt},  (8.8)

with initial data π^{ε}(0, ·) = π_{0}, where L_{ε}, Λ_{ε}, Ψ_{ε}, Θ_{ε}, β_{εt} are defined in the same way as L, Λ, Ψ, Θ, β_{t} with y_{t}^{ε} in place of y_{t}, and β̄_{εt} := (π^{ε}(t, ·), β_{ε}(t, ·, y_{t}^{ε})).

By Theorem 5.1, we have ||π^{ε}||_{H_{p}^{2}(T)} ≤ N for any T, where N is independent of ε. By Theorem 3.11 there is a π^{0} ∈ ∩_{T >0}H_{p}^{2}(T) and a sequence ε_{n} → 0 such that, for any ξ ∈ L_{q}(Ω) with q = p/(p − 1) and ϕ ∈ C_{0}^{∞} and t ≥ 0,

lim_{n→∞} E(π^{ε_{n}}(t, ·), ϕ)ξ = E(π^{0}(t, ·), ϕ)ξ.  (8.9)

This equality and the fact that (π^{ε_{n}}(t, ·), ϕ) = E{ϕ(x_{t}^{ε_{n}})|F_{t}^{ε_{n}}} (a.s.) can be used instead of equality (3.14) of [24]. As there, first for ξ of type g(y_{t_{1}}, ..., y_{t_{m}}) and then for arbitrary F_{t}^{y}–measurable ξ ∈ L_{q}(Ω), we get that

Eξ(π^{0}(t, ·), ϕ) = EξE{ϕ(x_{t})|F_{t}^{y}}.

If we knew that π^{0}(t, ·) is F_{t}^{y}–measurable, then this equality would imply that

(π^{0}(t, ·), ϕ) = E{ϕ(x_{t})|F_{t}^{y}} (a.s.)  (8.10)

which is nothing but assertion that the conditional distribution of x_{t} given F_{t}^{y} has a density coinciding with π^{0}(t, ·). Since π^{0} ∈ ∩_{T >0}H_{p}^{2}(T), equality (8.10) would imply that π^{0} satisfies (8.2) in the same way as at the beginning of the proof. Therefore, in the remaining part of the proof we only need to prove the F_{t}^{y}–measurability of π^{0}(t, ·).

By Itô’s formula and (8.8), the function

π̄^{ε}(t, x) := π^{ε}(t, x)ρ_{t}^{ε},  (8.11)

ρ_{t}^{ε} := exp\{∫_{0}^{t}(Θ_{ε}^{∗}Ψ_{ε}β̄_{εs}, dw_{s}) + ∫_{0}^{t}(β̄_{εs}, β_{εs}) ds − \frac{1}{2} ∫_{0}^{t} |β̄_{εs}|^{2} ds\}

satisfies the linear equation

dπ̄^{ε} = L_{ε}π̄^{ε} dt + ∑_{k≤d_{1}−d} Λ_{ε}^{k}π̄^{ε} {Ψ_{ε}^{k}Θ_{ε} dw_{t} + β_{εt}^{k} dt},

with initial data π̄^{ε}(0, ·) = π_{0}^{ε}. By using that sup_{t≤T} |y_{t}^{ε} − y_{t}| → 0 in probability and by applying Theorem 5.7 with n = −1 (here we also use Remark 5.8 and the continuity of θ_{x} with respect to y ), one easily proves that π̄^{ε} → π̄ in H_{p}^{1}(T) for any T, where π̄ satisfies (8.3). By Lemma 8.2 we have that π̄(t, ·) is F_{t}^{y}–adapted.

Next, by using the fact that π^{ε}(t, x) is a probability density, one gets from (8.11) that (cf. [24])

ρ_{t}^{ε} → ∫_{ℝ^{d}} π̄(t, x) dx =: ρ̄_{t}

in probability and the last term is again F_{t}^{y}–adapted. The process ρ̄_{t} satisfies the equation obtained by integrating (8.3), which means that

dρ̄_{t} = ρ̄_{t}β̂_{t} dy_{t},  (8.12)

where

β̂_{t} = ∫_{ℝ^{d}} β(t, x)π̄(t, x) dx(∫_{ℝ^{d}} π̄(t, x) dx)^{−1} (0 · 0^{−1} := 0).

It follows from (8.12) and the boundedness of β and β̂_{t} that ρ̄_{t} ≠ 0 with probability one.

Thus, for any ϕ ∈ C_{0}^{∞},

(π^{ε}(t, ·)ρ_{t}^{ε}, ϕ) → (π̄(t, ·), ϕ),

(π^{ε}(t, ·), ϕ) = \frac{(π^{ε}(t, ·)ρ_{t}^{ε}, ϕ)}{ρ_{t}^{ε}} → (π̄(t, ·), ϕ)(∫_{ℝ^{d}} π̄(t, x) dx)^{−1}

in probability. This implies that

(π^{0}(t, ·), ϕ) = (π̄(t, ·), ϕ)(∫_{ℝ^{d}} π̄(t, x) dx)^{−1} (a.s.),

where the last expression is F_{t}^{y}–adapted. The theorem is proved.

Remark 8.3. This theorem, along with embedding theorems from Sec. 7, extends and generalizes the corresponding results from [24] and [30] in the way we were talking about in the Introduction.

8.2. On the Notion of Stochastic Integral. To consider applications to equations with infinitely dimensional Wiener processes, we want to discuss the notion of stochastic integral and show that “basically” there is nothing more general than series of usual one– dimensional stochastic integrals. This will show that equations like (2.1), containing series of integrals with respect to Wiener processes, are of a quite general nature.

The first notion was introduced by Paley, Wiener, and Zygmund in [29], where the stochastic integral of a nonrandom smooth function f(t) against a one-dimensional Wiener process w_{t} is defined as

∫_{0}^{1} f(s) dw_{s} = f(1)w_{1} − ∫_{0}^{1} w_{s}f′(s) ds.  (8.13)

Then it is verified that the L_{2}(Ω)–norm of the stochastic integral is equal to the L_{2}(0, 1)– norm of f, which allows one to extend the stochastic integral from smooth functions to all f ∈ L_{2}(0, 1). One obtains the same integral if, instead of (8.13), one starts with

∫_{0}^{1} f(s) dw_{s} = ∑_{i=1}^{n} a_{i}(w_{s_{i}} − w_{s_{i−1}})  (8.14)

for functions f such that f(t) = a_{i} on (s_{i−1}, s_{i}] and 0 = s_{0} ≤ s_{1} ≤ ... ≤ s_{n} = 1.

The definition based on (8.14) has an advantage that it can be easily generalized to define an integral of a nonrandom function against a random orthogonal measure on a σ–finite measure space (X, X , m). More precisely, assume that we are given a π–system Π of subsets of X such that σ(Π) = X, and a random (complex–valued) variable μ(γ) defined for each γ ∈ Π (and perhaps not for all γ ∈ X). Assume that μ(γ) ∈ L_{2}(Ω) and Eμ(γ_{1})μ̄(γ_{2}) = m(γ_{1} ∩ γ_{2}) for each γ, γ_{1}, γ_{2} ∈ Π. Then for functions

f(x) = ∑_{i=1}^{n} a_{i}I_{γ_{i}}(x),

where a_{i}s are constant and γ_{i} ∈ Π, one defines the stochastic integral of f against μ as

∫_{X} f(x) μ(dx) = ∑_{i=1}^{n} a_{i}μ(γ_{i}),  (8.15)

and, again by isometry, one extends the stochastic integral to all f ∈ L_{2}(X, m). Such integrals are used in the theory of stationary processes. Surprisingly enough, as we will see, one can also say that this is the most general stochastic integral in Itô’s stochastic calculus.

Another advantage of (8.14) is that one can allow f to depend on ω, and if a_{i} are independent of the process w_{t+s_{i−1}} − w_{s_{i−1}}, t ≥ 0, then (8.14) is again an isometry between a part of L_{2}(Ω × (0, 1]) and a part of L_{2}(Ω). Closing this isometry, K. Itô defines his famous integral.

It turns out that Itô’s integral is a particular case of the integral based on (8.15). To be more precise, let w_{t} be (as usual) a Wiener process with respect to a filtration F_{t}, P be the predictable σ–field on Ω × (0, 1], and Π be the set of all stochastic intervals ^{|}(0, τ]], where τ are stopping times ≤ 1. Then one gets Itô’s integral by taking X = Ω × (0, 1] , X = P, μ^{|}(0, τ]] = w_{τ} (see more about this in [19]). In the same way one defines stochastic integrals with respect to any locally square integrable martingale.

K. Itô [12] was also the first to consider integration against measure–valued processes, which is a particular case of integration against martingale measures. Let p(t, Γ), t ≥ 0, be a square integrable process as a function of t with independent increments in time and a random orthogonal measures as a function of Γ for any t. Define p((s, t], Γ) = p(t, Γ) − p(s, Γ). Itô’s way of introducing the integral with respect to p is to replace the expression a_{i}(w_{s_{i}} − w_{s_{i−1}}) in (8.14) with

∫_{X} f_{i}(x) p((s_{i−1}, s_{i}], dx) = ∫_{X} f_{i}(x) p(s_{i}, dx) − ∫_{X} f_{i}(x) p(s_{i−1}, dx),  (8.16)

where f_{i} are assumed to be independent of the processes p((s_{i−1}, t], Γ), t ≥ s_{i−1}.

More generally, for any γ ∈ Π let a process p(t, γ) be given, which is a square integrable martingale with respect to a given filtration {F_{t}}. Let ⟨p(·, γ_{1}), p(·, γ_{2})⟩_{t} = q(t, γ_{1} ∩ γ_{2}), where q(t, ·) is a σ–finite measure on (X, X) for any ω, t, and q(t, Γ) increases in t for any Γ ∈ X and ω. Then there is a measure q(dt, dx) such that

q(t, γ) = ∫_{0}^{1} ∫_{γ} q(ds, dx).

By following Itô’s method based on (8.16), for any P × X –measurable f = f(ω, t, x) such that

∫_{0}^{1} ∫_{X} f^{2}(s, x) q(ds, dx) < ∞,

one defines the stochastic integral

∫_{0}^{1} ∫_{X} f(s, x) p(ds, dx).  (8.17)

This integral is also a particular case of the integral of a nonrandom function against a random orthogonal measure. Indeed, define X̄ = Ω × (0, 1] × X, X̄ = P × X, and let m(dωdtdx) := P (dω)q(dt, dx). Also let

Π̄ = {^{|}(0, τ]] × γ : Eq(^{|}(0, τ]] × γ) < ∞}, μ(^{|}(0, τ]] × γ) = p(τ, γ).

Then on functions

f = ∑_{i≤n} a_{i}I_{t≤τ_{i}}I_{γ_{i}}(x),  (8.18)

where ^{|}(0, τ_{i}]] × γ_{i} ∈ Π̄ and a_{i} are some constants, integral (8.17) equals

∑_{i≤n} a_{i}p(τ_{i}, γ_{i}) = ∑_{i≤n} a_{i}μ(^{|}(0, τ_{i}]] × γ_{i}),

which agrees with (8.15). Finally, by functions of type (8.18) one can approximate any P × X –measurable function for which (8.17) can be defined.

It is worth mentioning that there are also other notions of martingale measures with respect to which one can define stochastic integration (see [35], where the martingale measures discussed above are called orthogonal martingale measures).

Even though the notion of integral of nonrandom functions with respect to random orthogonal measures is very convenient for the purpose of introducing Itô’s stochastic integrals (cf. [19]), one works almost always with stochastic integrals with variable limits, and a different notation is more appropriate.

In connection with this notice that it is shown in [10] how to reduce the stochastic integral with respect to a martingale measure to a series of usual stochastic integrals. This was further used in [10] to treat stochastic equations containing integrals against martingale measures using the same notation as in the case of equations containing just usual stochastic integrals.

To be more precise, it is assumed in [10] that X is countably generated and

q(t, Γ) = ∫_{0}^{t} q_{s}(Γ) dV_{s},

where V_{s} is a predictable increasing process and q_{s}(Γ) is a measure in Γ for any s and predictable in s for any Γ ∈ X. Then it is shown that

∫_{0}^{t} ∫_{X} f(s, x) p(ds, dx) = ∑_{k=1}^{∞} ∫_{0}^{t} f_{k}(s) dp_{s}^{k},  (8.19)

where

p_{t}^{k} = ∫_{0}^{t} ∫_{X} η_{k}(s, x) p(ds, dx), f_{k}(s) = ∫_{X} η_{k}(s, x)f(s, x) q_{s}(dx),

and for any ω, s the system of functions {η_{k}(s, ·)} forms an orthonormal basis in L_{2}(X, q_{s}).

A particular case of the stochastic integral with respect to a martingale measure is the stochastic integral with respect to the two–dimensional Brownian sheet W (t, x) defined for t ≥ 0, x ∈ ℝ. In this case, one takes F_{t} so that the random variables W (t, x) are F_{t}–measurable, and defines p(^{|}(0, τ]] × (a, b]) = W (τ, b) − W (τ, a). This integral got very popular thanks to the article [35]. One can construct W (s, x) by taking independent one– dimensional Wiener processes w_{t}^{k}, k ≥ 1, and an orthonormal basis {η_{k}(x), k ≥ 1} in L_{2}(ℝ), and letting

W (t, x) = ∑_{k=1}^{∞} w_{t}^{k} ∫_{0}^{x} η_{k}(y) dy t ≥ 0, x ∈ ℝ.

Incidentally, observe that thus defined W is a Gaussian field and

EW (s, y)W (t, x) = s ∧ t ∑_{k=1}^{∞} ∫_{0}^{x} η_{k}(z) dz ∫_{0}^{y} η_{k}(z) dz,

where

∑_{k=1}^{∞} ∫_{0}^{x} η_{k}(z) dz ∫_{0}^{y} η_{k}(z) dz = \{ \begin{matrix}∫_{ℝ} I_{(0,x)}(z)I_{(0,y)}(z) dz = x ∧ y & if x, y ≥ 0, \\ ∫_{ℝ} I_{(x,0)}(z)I_{(y,0)}(z) dz = |x| ∧ |y| & if x, y ≤ 0, \\ 0 & otherwise. \end{matrix}

In this particular case (8.19) becomes

∫_{0}^{t} ∫_{ℝ} f(s, x) W (ds, dx) = ∑_{k=1}^{∞} ∫_{0}^{t} {∫_{ℝ} η_{k}(x)f(s, x) dx} dw_{s}^{k}.  (8.20)

By the way, general one–dimensional equations driven by the cylindrical space–time white noise B^{.}_{t} were considered in [7], where the right–hand side of (8.20) is taken by definition as ∫_{0}^{t}⟨f(s, ·), dB_{t}⟩. There the series was introduced from the very beginning.

The last integral we want to discuss is the integral against a Hilbert–space valued Wiener processes (see, for instance, [30]). Let H be a Hilbert space and w_{t} be a H–valued Wiener process with covariance operator Q. This operator is known to be nuclear. If h_{k} are its unit eigenvectors with nonzero eigenvalues, then w_{t}^{k} := (h^{k}, w_{t})(Qh^{k}, h^{k})^{−1/2} are independent standard Wiener processes and, for any H–valued process f_{t} for which the integral ∫_{0}^{t} f_{t} ·dw_{t} is defined, the integral can be written as

∑_{k} ∫_{0}^{t} f_{s}^{k} dw_{s}^{k},

where f_{s}^{k} = (f_{s}, h^{k})(Qh^{k}, h^{k})^{1/2} (see, for instance, [30]).

8.3. Equations Driven by Space–Time White Noise. In this subsection, we consider one-dimensional equations with space-time white noise. Thus d = 1. Very often (see, for instance [7]) one writes these equations in the form

du(t, x) = [a(t, x)u′′(t, x) + b(t, x)u′(t, x) + f(t, x, u(t, x))] dt + h(t, x, u(t, x)) dB_{t}, (8.21)

where B_{t} is a cylindrical Wiener process on L_{2}. There are also very many articles (see, for instance [8]) where instead of dB_{t} one writes (∂^{2}W/∂t∂x) dt. As we explained in Subsec. 8.2 we may as well take ∑_{k} η_{k}(x) dw_{t}^{k}, where {η_{k}(x), k ≥ 1} is an orthonormal basis in L_{2} and w_{t}^{k} are independent F_{t}–adapted one–dimensional Wiener processes. Thus, instead of (8.21), we will be considering the equation

\begin{matrix}du(t, x) = & [a(t, x)u′′(t, x) + b(t, x)u′(t, x) + f(t, x, u(t, x))] dt \\ + g^{k}(t, x, u(t, x)) dw_{t}^{k} \end{matrix}  (8.22)

on a time interval [0, τ], where g^{k} := hη_{k} and τ is a bounded stopping time.

Assumption 8.5. The functions a(t, x) = a(ω, t, x) and b(t, x) = b(ω, t, x) are realvalued functions defined on ^{|}(0, τ]] × ℝ.

(i) For any ω and t ≤ τ(ω), a(ω, t, ·) ∈ C^{1,1} (= B^{2}) and b(ω, t, ·) ∈ C^{0,1} (= B^{1}) and ||a||_{C^{1,1}} + ||b||_{C^{0,1}} ≤ K. Also K ≥ a ≥ δ.

(ii) For any x ∈ ℝ, the processes a and b are predictable.

To state the next assumption, take and fix s ≤ ∞ and finite κ, p, r such that

κ ∈ (0, 1/2], p ≥ 2r ≥ 2, s ≤ ∞, \frac{1}{r} + \frac{1}{s} = 1, r < \frac{1}{1 − 2κ}.  (8.23)

Assumption 8.6. The functions f(t, x, u) and h(t, x, u) are real-valued functions on |(0, τ]] × ℝ^{2} such that

(i) for any x and u, the processes f(t, x, u) and h(t, x, u) are predictable;

(ii) for any ω, t, x, u, and v,

|f(t, x, u) − f(t, x, v)| ≤ K|u − v|, |h(t, x, u) − h(t, x, v)| ≤ ξ(t, x)|u − v|,  (8.24)

where ξ is certain function of ω, t, x satisfying ||ξ(t, ·)||_{2s} ≤ K.

Observe that one of possibilities is r = 1, and then s = ∞ and (8.24) just means that both f and h satisfy the Lipschitz condition in u with constant K.

To fit equation (8.22) in our general scheme, we need to find an appropriate n such that the assumptions of Theorem 5.1 are satisfied. Define

n = −κ − 3/2.

In the following lemma, we also set

R(x) = χ|x|^{−(1−2κ)/2} ∫_{0}^{∞} t^{−(5−2κ)/4}e^{−tx^{2}−1/(4t)} dt,

so that, according to (2.3) and (2.4) for right choice of the constant χ, the function R(x) is the kernel of the operator (1−Δ)^{(n+1)/2}. It is easy to prove that R is infinitely differentiable everywhere except the origin, decreases exponentially fast as |x| → ∞, and behaves near the origin like |x|^{−(1−2κ)/2} if κ < (1/2) and like − log |x| if κ = 1/2. Finally, notice that generally speaking, the notation h, ξ, g is used in the lemma for functions different from the ones introduced above.

Lemma 8.4. Take some functions h ∈ L_{p}, ξ ∈ L_{2s}, and set g^{k} = ξhη_{k}. Then g = {g^{k}}_{k≥1} ∈ H_{p}^{n+1}(l_{2}) and

||g||_{n+1,p} = ||h̄||_{p} ≤ (N||ξ||_{2s}||h||_{p}) ∧ (||ξh||_{2}||R||_{p}),  (8.25)

where N = ||R||_{2r} < ∞ and

h̄(x) := {∫_{ℝ} R^{2}(x − y)ξ^{2}(y)h^{2}(y) dy}^{1/2} .  (8.26)

In addition, if p(1 − 2κ) > 2 and ξ = 1, then

||g||_{n+1,p} ≤ N(κ)||h||_{2}^{2κp/(p−2)}||h||_{p}^{1−2κp/(p−2)}.  (8.27)

Proof. We know that

(1 − Δ)^{(n+1)/2}(ξhη_{k})(x) = ∫_{ℝ} R(x − y)ξ(y)h(y)η_{k}(y) dy.

It follows that by Parseval’s theorem

|(1 − Δ)^{(n+1)/2}g(x)|_{l_{2}}^{2} = ∑_{k=1}^{∞} (∫_{ℝ} R(x − y)ξ(y)h(y)η_{k}(y) dy)^{2}

= ∫_{ℝ} R^{2}(x − y)ξ^{2}(y)h^{2}(y) dy = h̄^{2}(x).

We thus get the equality in (8.25). To prove the inequality, notice that h̄^{2} is a convolution and by Minkowski’s inequality the L_{p}–norm of a convolution is less than the L_{1}–norm of one function times L_{p}–norm of another. This immediately gives ||h̄||_{p} ≤ ||ξh||_{2}||R||_{p}. Also, by Hölder’s inequality,

h̄^{2}(x) ≤ ||ξ||_{2s}^{2} (∫_{ℝ} R^{2r}(x − y)h^{2r}(y) dy)^{1/r} ,

which leads to the second inequality in (8.25): ||h̄||_{p} ≤ N||ξ||_{2s}||h||_{p}, again by Minkowski’s inequality and by the assumption p ≥ 2r. The finiteness of N follows from the assumption that r < (1 − 2κ)^{−1}.

To prove (8.27), we use that R^{2}(y) ≤ N|y|^{2κ−1} and we minimize with respect to ε > 0 after the following computations:

(∫_{ℝ}{∫_{ℝ} R^{2}(y)h^{2}(x − y) dy}^{p/2} dx)^{2/p} ≤ (∫_{ℝ}{∫_{ℝ} I_{|y|≤ε}R^{2}(y)h^{2}(x − y) dy}^{p/2} dx)^{2/p}

+(∫_{ℝ}{∫_{ℝ} I_{|y|≥ε}R^{2}(y)h^{2}(x − y) dy}^{p/2} dx)^{2/p} ≤ ||I_{|y|≤ε}R^{2}||_{1}||h||_{p}^{2} + ||I_{|y|≥ε}R^{p}||_{1}^{2/p}||h||_{2}^{2}

≤ Nε^{2κ}||h||_{p}^{2} + Nε^{−(1−2κ−2/p)}||h||_{2}^{2}.

The lemma is proved.

Theorem 8.5. Let κ ∈ (0, 1/2) and u_{0} ∈ L_{p}(Ω, F_{0}, H_{p}^{(1/2)−κ−2/p}). Assume that
I^{p}(τ) := E ∫_{0}^{τ} {||f(t, ·, 0)||_{−3/2−κ,p}^{p} + ||h̄(t, ·, 0)||_{p}^{p}} dt < ∞,  (8.28)

where

h̄(t, x, 0) := {∫_{ℝ} R^{2}(x − y)h^{2}(t, y, 0) dy}^{1/2} .

Then, in the space H_{p}^{(1/2)−κ}(τ), equation (8.22) with the initial condition u_{0} has a unique solution u. Moreover,

||u||_{H_{p}^{1/2−κ}(τ)} ≤ N{I(τ) + (E||u_{0}||_{1/2−κ−2/p,p}^{p})^{1/p}},

where the constant N depends only on κ, p, δ, K, and τ.

Proof. We will apply Theorem 5.1. Its assumptions concerning a and σ (≡ 0) are obviously satisfied. Next, if u ∈ H_{p}^{n+2}, then bu′ ∈ H_{p}^{n+1} ⊂ H_{p}^{n}, u ∈ L_{p}, f(u) − f(0) ∈ L_{p} ⊂ H_{p}^{n}. Also, by (8.28) we have f(t, ·, 0) ∈ H_{p}^{n} (a.e. on ^{|}(0, τ]]). Furthermore, by Lemma 5.2,

||bu′||_{n,p} ≤ ||bu′||_{−1,p} ≤ N||u′||_{−1,p} ≤ N||u||_{p} = N||u||_{n+2−((1/2)−κ),p},
||f(u) − f(v)||_{n,p} ≤ ||f(u) − f(v)||_{p} ≤ K||u − v||_{p}.

We emphasize that || · ||_{p} = || · ||_{n+2−((1/2)−κ),p} and n + 2 − ((1/2) − κ) < n + 2 for κ < 1/2. Consequently (see Remark 5.5), the assumptions of Theorem 5.1 concerning bu′ + f(u) are satisfied. To check the remaining assumptions about g(u), it suffices to notice that, by Lemma 8.4, we have

||g(0)(t, ·)||_{n+1,p} = ||h̄(t, ·, 0)||_{p}, ||g(u)(t, ·) − g(v)(t, ·)||_{n+1,p}
≤ N||h(t, ·, u(t, ·)) − h(t, ·, v(t, ·))||_{p} ≤ N||u(t, ·) − v(t, ·)||_{p}.

The theorem is proved.

Remark 8.6. We have obtained Theorem 8.5 by checking that all Assumptions 5.1– 5.6 are satisfied for n = −3/2 − κ. After this, of course, all other results from Sec. 5 are available. For example, by the approximation theorem (Theorem 5.7) and Remark 5.9, we have ||u − u_{m}||_{H_{p}^{n+2}(τ)} → 0, where u_{m} is a unique solution of the following version of (5.11)

du(t, x) = [a(t, x)u′′(t, x) + b(t, x)u′(t, x) + f(t, x, u(t, x))] dt + ∑_{k=1}^{m} h(t, x, u(t, x))η_{k} dw_{t}^{k}

with initial condition u_{0}.

Remark 8.7. Additional information about Hölder continuity properties of the solution is readily obtained from the properties of elements of H_{p}^{n}(τ) listed in Theorem 7.2.

For example,

u ∈ C^{α−1/p}([0, τ], H_{p}^{n+2−2β})  (a.s.)  (8.29)

provided p > 2 and 1/2 > β > α > 1/p. Here H_{p}^{n+2−2β} ⊂ C^{γ} if

γ := n + 2 − 2β − 1/p = 1/2 − κ − 2β − 1/p > 0.

Hence, if the inequality E||u_{0}||_{1/2−κ−2/p,p}^{p} < ∞ and (8.28) are satisfied for any κ ∈ (0, 1/2) and p ≥ 2 (say f(t, x, 0) = h(t, x, 0) = 0, ξ = K, and u_{0} is a deterministic smooth function with compact support), then, after taking p large enough and κ, α, and β small, we see that u satisfies the Hölder condition in x of order 1/2 − ε uniformly with respect to t ∈ [0, τ] (a.s.) for any ε > 0.

On the other hand, by taking p large enough, both α and β close to 1/4, and κ small, we get that u satisfies the Hölder condition in t of order 1/4 − ε uniformly with respect to x ∈ ℝ (a.s.) for any ε > 0. In terms of parabolic Hölder spaces, this means that

u ∈ C_{t , x}^{1/4−ε,1/2−ε}([0, τ] × ℝ) (a.s.).

This result agrees well with [35], where the same continuity is obtained for equations with constant coefficients. Also, notice that, using standard PDE methods, one can prove interior (with respect to t) estimates which would give similar continuity of u away from t = 0 under weaker assumptions on u_{0}. One of results which can be obtained is discussed in the following remark.

Remark 8.8. There are smoothing properties of equations in the sense that solutions may be much smoother than the initial data. For example, assume that f(t, x, 0) = h(t, x, 0) = 0, τ = T, where T is a constant, r = 1, and s = ∞. Let u_{0} ∈ L_{2}(Ω, F_{0}, H_{2}^{−1+ε}) with ε ∈ (0, 1/2), so that u_{0} may be a finite measure or just a delta–function. We claim then that, for any ε ∈ (0, 1/4),

u ∈ C_{t , x}^{1/4−ε,1/2−ε}([ε, T] × ℝ) (a.s.).

Indeed, by Theorem 8.5 with p = 2 and κ = 1/2−ε, there is a unique solution u ∈ H_{2}^{ε}(T) of (8.22) and

||u||_{H_{2}^{ε}(T)}^{2} ≤ NE||u_{0}||_{−1/2−κ,2}^{2}.

By (3.4) we have

∫_{0}^{T} E||u(t, ·)||_{ε,2}^{2} dt = ||u||_{ℍ_{2}^{ε}(T)}^{2} ≤ NE||u_{0}||_{−1/2−κ,2}^{2} =: I.

By Chebyshev’s inequality, for any γ ∈ (0, T/2), Lebesgue measure of the set on (γ, 2γ), where E||u(t, ·)||_{ε,2}^{2} > 2I/γ, is less than γ/2. This implies that for any γ ∈ (0, T/2) there is a point t_{γ} ∈ (γ, 2γ) such that

u(t_{γ}, ·) ∈ L_{2}(Ω, F_{t_{γ}} , H_{2}^{ε}), E||u(t_{γ}, ·)||_{ε,2}^{2} ≤ 2I/γ.

By considering (8.22) after time t_{γ} instead of 0 and defining the spaces H_{2}^{1/2−κ}(t_{γ}, T) in an obvious way, we get by Theorem 8.5 that u ∈ H_{2}^{1/2−κ}(t_{γ}, T) for any κ ∈ (0, 1/2). In addition,

∫_{t_{γ}}^{T} E||u(t, ·)||_{1/2−κ,2}^{2} dt ≤ NI/γ, E||u(s_{γ}, ·)||_{1/2−κ,2}^{2} ≤ NI/γ^{2},  (8.30)

where s_{γ} is a certain point in (t_{γ} + γ, t_{γ} + 2γ) and γ ≤ T/4.

Now we can go to larger powers. Take any p > 4 and define A(p, γ, R) = {ω : ||u(s_{γ}, ·)||_{p} ≤ R}. By Sobolev’s embedding theorem, H_{2}^{1/2−κ} ⊂ H_{p}^{m} if m < 1/2 − κ and −κ = m − 1/p. For m = 0 we have κ = 1/p. Since in (8.30) we can take κ = 1/p, we get

P {A(p, γ, R)} = P {||u(s_{γ}, ·)||_{p} ≤ R} ≥ P {||u(s_{γ}, ·)||_{1/2−κ,2} ≤ NR} ≥ 1 − NIγ^{−2}R^{−2}.

Next, obviously, on the set A(p, γ, R) and the time interval (s_{γ}, T), the assumptions of Theorem 8.5 are satisfied with κ = 1/2 − 2/p ∈ (0, 1/2). Therefore

∫_{s_{γ}}^{T} EI_{A(p,γ,R)}||u(t, ·)||_{2/p,p}^{p} dt ≤ NEI_{A(p,γ,R)}||u(s_{γ}, ·)||_{p}^{p} ≤ NR^{p},
EI_{A(p,γ,R)}||u(r_{γ}^{1}, ·)||_{2/p,p}^{p} ≤ NR^{p}/γ,

for an r_{γ}^{1} ∈ (s_{γ} + γ, s_{γ} + 2γ) and γ < T/6.

For p > 8, we can represent 2/p as 1/2 − κ − 2/p with κ = 1/2 − 4/p ∈ (0, 1/2). For those p, by Theorem 8.5,

∫_{r_{γ}^{1}}^{T} EI_{A(p,γ,R)}||u(t, ·)||_{4/p,p}^{p} dt ≤ NEI_{A(p,γ,R)}||u(r_{γ}^{1}, ·)||_{2/p,p}^{p} ≤ NR^{p}/γ,
EI_{A(p,γ,R)}||u(r_{γ}^{2}, ·)||_{4/p,p}^{p} ≤ NR^{p}/γ^{2},

for some r_{γ}^{2} ∈ (r_{γ}^{1} + γ, r_{γ}^{1} + 2γ) ∈ (0, T). One can keep going this way and, for p > 4n, where n = 1, 2, ..., find r_{γ}^{n} that are close to zero if γ is small, such that I_{A(p,γ,R)}u ∈ H_{p}^{2n/p}(r_{γ}^{n}, T). Here p can be taken arbitrary large, r_{γ}^{n} small, and 2n/p can be made as close to 1/2 as we wish. Also, the probability of the set A(p, γ, R) can be chosen close to 1. Therefore, we obtain our claim as in Remark 8.7.

Remark 8.9. If ε ≥ 0 and κ + 2ε < 1/2, then in (8.21) we could take a noise like (1 − Δ)^{ε}B_{t}, which is even “whiter” than B_{t}. This would only lead to replacing η_{k} in (8.22) with (1 − Δ)^{ε}η_{k}, and, under natural additional smoothness assumptions on h, would give the assertions of Theorem 8.5 with κ + 2ε in place of κ.

8.4. Non–Explosion for a Nonlinear Equation. Take a, b, h satisfying Assumptions 8.5 and 8.6 from Subsec. 8.3 for τ = ∞, r = 1, s = ∞, and ξ ≡ 1 and take a bounded real-valued P × B(ℝ)-measurable function c(t, x) = c(ω, t, x). Assume that h(t, x, u) = 0 for u ≤ 0. Fix a number λ ∈ [0, 1/2) and let

g^{k}(t, x, u) = h(t, x, u)u_{+}^{λ}η_{k}(x),

where, as usual, {η_{k}} form an orthonormal basis in L_{2}. Here it is convenient to assume additionally that each η_{k} is bounded.

The results from Subsec. 8.3 can be easily applied to prove that if the initial condition u_{0} is nonnegative and, say, is nonrandom and belongs to C_{0}^{∞}, then the equation

\begin{matrix}du(t, x) = & [au′′(t, x) + b(t, x)u′(t, x) + c(t, x)u(t, x)] dt \\ + g^{k}(t, x, u(t, x)) dw_{t}^{k} \end{matrix}  (8.31)

has a solution defined for all t in the class of functions such that sup_{t≤T,x} |u(t, x)| is finite (a.s.) for any T < ∞. These facts for equation (8.31) considered on a finite space interval with a ≡ 1, b ≡ c ≡ 0, and h(u) = u_{+} and with zero boundary data were discovered in [26] with the help of a quite different approach. By using the maximum principle, one can show that our assertion implies the result of [26].

First, let us explain why (8.31) is solvable despite the high growth of g in u. It turns out that (8.31) possesses a kind of integral or conservation law. Observe that u ≥ 0, which follows from the maximum principle if one notices that the solution u of (8.31) also satisfies the equation with g^{k}(t, x, u(t, x)) replaced by ν^{k}(t, x)u(t, x), where ν^{k}(t, x) = g^{k}(t, x, u(t, x))u^{−1}(t, x) and |ν^{k}| ≤ N|u|^{λ}. Moreover, if, for instance, a = 1, b = c = 0, then, by integrating (8.31) formally with respect to x, one obtains that ||u(t, ·)||_{1} is a local martingale. It is nonnegative, therefore its trajectories are bounded (a.s.).

This takes care of “almost u^{1/2}” in the diffusion term. Indeed, one can rewrite (8.31) with ξ(t, x)uη_{k}(x) in place of g^{k}(t, x, u), where ξ(t, x) = h(t, x, u(t, x))u^{λ−1}(t, x) and |ξ(t, x)| ≤ |u^{λ}(t, x)|. By the above, the latter is summable to power 2s = 1/λ, and s > 1 (which is required in Theorem 8.5) for λ < 1/2. Therefore, u satisfies a linear equation with coefficients under control, and we get that u(t, x) is a bounded continuous function on [0, T] × ℝ for any T < ∞ as in Remark 8.7 (by letting p → ∞). The rigorous treatment below follows this idea.

By Theorem 8.5, for any m = 1, 2, 3..., the equation

du_{m} = (au′′_{m} + bu′_{m} + cu_{m}) dt + g^{k}(u_{m} ∧ m) dw_{t}^{k}  (8.32)

with the initial condition u_{m}(0, ·) = u_{0} has a unique solution u_{m} ∈ H_{p}^{1/2−κ}(T) for any κ ∈ (0, 1/2), p ≥ 2, and T < ∞. By Remark 8.7, the function u_{m}(t, x) is continuous in (t, x) (a.s.). To proceed with the argument, we need the following lemma to be proved later.

Lemma 8.10. For any T < ∞,

lim_{R→∞} sup_{m} P { sup_{t≤T,x} |u_{m}(t, x)| ≥ R} = 0.  (8.33)

Define

τ_{m}^{R} = inf{t ≥ 0 : sup_{x} |u_{m}(t, x)| ≥ R}.

Observe that if m ≥ R (and R is an integer), then the functions u_{m} and u_{R} satisfy the same equation on [0, τ_{m}^{R}) and therefore coincide by uniqueness. In particular, τ_{m}^{m} ≥ τ_{m}^{R} = τ_{R}^{R} (a.s.). Therefore, there is no ambiguity in the definition

u(t, x) = u_{m}(t, x) on [0, τ_{m}^{m}].

Of course, u satisfies (8.31) on (0, lim τ_{m}^{m}) and |u(t, x)| ≤ m for t ≤ τ_{m}^{m}. To finish the proof of our assertions about (8.31), it only remains to notice that lim τ_{m}^{m} = ∞ (a.s.), since, by (8.33),

P {τ_{m}^{m} ≤ T } = P { sup_{t≤T,x} |u_{m}(t, x)| ≥ m} ≤ sup_{n} P { sup_{t≤T,x} |u_{n}(t, x)| ≥ m} → 0

as m → ∞. Proof of Lemma 8.10. Define

ξ_{m}(t, x) = h(t, x, u_{m}(t, x) ∧ m)(u_{m}(t, x) ∧ m)_{+}^{λ}u_{m}^{−1}(t, x) (0 · 0^{−1} := 0)

and notice that ξ_{m} is a bounded function. It follows from (8.32) that u_{m} is a solution of the equation

dv = (av′′ + bv′ + cv) dt + vξ_{m}η_{k} dw_{t}^{k}.  (8.34)

Obviously, Assumptions 8.5 and 8.6 from Subsec. 8.3 are satisfied for (8.34). By Remark 8.6 and by virtue of our assumption about boundedness of η_{k}, Theorem 5.12 is valid for (8.34). Thus, u_{m} ≥ 0 (a.s.).

Next, take ζ_{k}(x) from Theorem 5.7, multiply (8.32) by ζ_{k}e^{−Kt}, where K = sup(|a′′| + |b′| + |c|), integrate by parts (that is, use the definition of solutions), and take expectations. Then, for any constant T and stopping time τ ≤ T, we obtain

e^{−KT} E(ζ_{k}, u_{m}(τ, ·)) ≤ E(ζ_{k}, u_{m}(τ, ·))e^{−Kτ} = (ζ_{k}, u_{0})
+E ∫_{0}^{τ} (aζ′′_{k} + (2a′ − b)ζ′_{k} + (a′′ − b′ + c − K)ζ_{k}, u_{m})e^{−Kt} dt

≤ N + NE ∫_{0}^{τ} (|ζ′′_{k}| + |ζ′_{k}|, u_{m}) dt ≤ N + \frac{N}{k} k^{1−1/p}E ∫_{0}^{T} ||u_{m}(t, ·)||_{p} dt,

E(ζ_{k}, u_{m}(τ, ·)) ≤ N + Nk^{−1/p}(E ∫_{0}^{T} ||u_{m}(t, ·)||_{p}^{p} dt)^{1/p} ≤ N + Mk^{−1/p}, where the last constant N is independent of m, k, and τ and M is independent of k. Since this inequality is true for any stopping time τ ≤ T, with the same N and M for any number γ ∈ (0, 1) (see, for instance, Theorem III.6.8 of [19]),

E sup_{t≤T}(∫_{ℝ} ζ_{k}(x)u_{m}(t, x) dx)^{γ} ≤ 1 + γ \frac{N + Mk^{−1/p}}{1 − γ} , E sup_{t≤T} ||u_{m}(t, ·)||_{1}^{γ} ≤ \frac{N + 1}{1 − γ},

where the latter relation is obtained from the former one by the monotone convergence theorem. It follows that

P \{sup_{t≤T} ||u_{m}(t, ·)||_{1} > S\} ≤ \frac{N}{\sqrt{S}} ,  (8.35)

where N is independent of m and S > 0.

Now fix m, S > 0 and define

τ_{m}(S) = inf{t ≥ 0 : ||u_{m}(t, ·)||_{1} ≥ S}, h_{m}(t, x, u) = ξ_{m}(t, x)u.

Observe that ξ_{m} ≤ u_{m}^{λ}, which yields that for t ≤ τ_{m}(S) we have ||ξ_{m}(t, ·)||_{2s} ≤ S^{1/2s} with s = 1/(2λ). Also, u_{m} satisfies (8.34). By Theorem 8.5, we obtain that, for any T < ∞, ||u_{m}||_{H_{p}^{1/2−κ}(T ∧τ_{m}(S))} is bounded by a constant independent of m, whenever p ≥ 2r = 2/(1 − 2λ) and 1/2 > κ > λ. The embedding theorems (cf. Remark 8.7) imply that E sup_{t≤T ∧τ_{m}(S),x} |u_{m}(t, x)|^{p} is bounded independently of m for all large p (and hence for small p as well).

Thus

P { sup_{t≤T,x} |u_{m}(t, x)| ≥ R} ≤ P { sup_{t≤T ∧τ_{m}(S),x} |u_{m}(t, x)| ≥ R} + P {τ_{m}(S) < T }

≤ \frac{N}{R} + P \{sup_{t≤T} ||u_{m}(t, ·)||_{1} > S\} ≤ \frac{N}{R} + \frac{N}{\sqrt{S}}

with the constants N independent of m, S, R. This leads to (8.33), and the lemma is proved.

Remark 8.11. In [24] one can find a proof that sup_{m} E||u_{m}||_{C([0,T]×ℝ)}^{γ} < ∞ for any T < ∞, where 0 < γ < 1 − 2λ and γ can be chosen arbitrary close to 1 − 2λ.

9. Open Problems

• In the filtering problem, we only considered the case of equation (8.1) with noise in observations depending only on t and y_{t}, and thus independent of the signal x_{t}. Almost nothing is known about filtering problems in which Θ depends on the signal x_{t}. Very interesting jump processes may appear.

• The above theory applies to equations in the whole space. It is known that even for zero Dirichlet boundary data the L_{2}–theory is much more complicated because one needs weighted Sobolev spaces (cf. [16]) or compatibility conditions (cf. [2], [6]). The L_{2}–theory in [16] was developed by considering the stochastic term as a perturbation, which gave a perfect result due to obtaining some sharp estimates which turned out to be usable. Nothing of that kind can happen for general p > 2, that is why very often, say in [6], it is assumed that the stochastic term is sufficiently small and some compatibility conditions hold. A first encouraging step toward developing L_{p}–theory in weighted Sobolev spaces in bounded domains is made in [21] (see also [23]).

By the way, the fact that the derivatives of solutions blow up near the boundary shows that one needs to be careful constructing finite–difference approximations near the boundary.

• The above theory is an L_{p}–theory. One may ask whether a C^{2+α}–theory can be constructed. By a theory we mean not only results that, for f, g^{k} belonging to a space F, the solution belongs to some kind of stochastic C^{2+α}–spaces, but also that every element of this stochastic space can be obtained as a solution for certain f, g^{k} belonging to the same F.

We only have Theorems 5.1 and 7.2 and Sobolev’s embedding H_{p}^{n} ⊂ C^{n−d/p} if n − d/p > 0. Thus, if f, g^{k} are relatively smooth, the solution belongs to C^{2+α}. This is a useful result; for instance, it essentially covers the main result of [3] obtained in the whole space for coefficients independent of x. The weakness of such results is that they do not allow natural perturbations and cannot be extended reasonably far. For example, if one knows such results only for equations with constant coefficients, one will not be able to extend them to equations with variable highest order coefficients.

However, a “right” C^{2+α} theory in the above sense is developed in [27], for equations with nonrandom coefficients and with no derivatives of the unknown function in the stochastic term.

• Fully nonlinear equations and uniqueness. In Assumption 5.6 we assume that nonlinear terms in equation (5.1) are strictly subordinated to the main terms. Nothing is known if, instead of linear main operators in (5.1), one has nonlinear ones. Even for the case of linear equations we do not know how strong this subordination has to be. For instance, take the following version of one–dimensional equation (8.22)

du(t, x) = u′′(t, x) dt + h(x)u(t, x)η^{k}(x) dw_{t}^{k},

where we only know that h ∈ L_{2}. Is it true that if sup_{t≤T,x} |u(t, x)| < ∞ (a.s.) and u(0, ·) = 0, then u = 0 on [0, T]? The point is that in Assumption 8.6 we have to take s = 1, and then r = ∞, p = ∞ and even κ = 1/2 is a little bit short of satisfying the last inequality in (8.23) which is needed to provide the right subordination. The author believes that the answer to this particular question is positive, but the general question still remains.

• Superdiffusions. One can derive SPDEs for superdiffusions in multi–dimensional case (see [22]). The question arises whether one can get from these equations at least some part of huge and delicate information about superprocesses, available in the literature.

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Index

(1 − Δ)^{n/2}, 186 ; (L, Λ), 190, 213 ; (u, ϕ), 190 ; (Ω, F, P), 190 ; (ϕ, u), 190 ; B(ℝ^{d}), 206 ; B^{|n|+γ}, 206 ; B^{|n|+γ}(l_{2}), 206 ; C^{α}, 218 ; C^{|n|+γ}(ℝ^{d}), 206 ; C^{|n|−1,1}(ℝ^{d}), 206 ; C_{0}^{∞}, 186 ; H_{p}^{1,2}(T), 187 ; H_{p}^{n}, 186 ; H_{p}^{n}(ℝ^{d}), 186 ; L(t, ·), 213 ; Lu, 186 ; L_{p}, 186 ; n_{b}, 208 ; n_{c}, 208 ; n_{ν}, 208 ; T_{t}, 187 ; W_{2}^{n}(ℝ^{d}), 185 ; W_{p}^{n}, 190 ; w_{t}^{k}, 186, 190 ; α^{ij}, 196, 207 ; ā, 207 ; 𝔻, 192 ; ℍ_{p}^{n}, 191 ; ℍ_{p}^{n}(τ), 191 ; ℍ_{p}^{n}(τ, l_{2}), 191 ; 𝕃_{...}..., 191 ; ℝ^{d}, 186 ; 𝕊, 192 ; C^{α}, 218 ; D, 186  F_{p}^{n}, 191 ; F_{p}^{n}(τ), 191 ; F_{t}, 190 ; H_{p}^{2}(T), 189 ; H_{p,0}^{n}(τ), 191 ; H_{p}^{n}, 191 ; H_{p}^{n}(τ), 191 ; R_{1}, 188 ; R_{λ_{0}}, 188, 218 ; γ, 206 ; ĥ_{m}, 211 ; Λu, 186 ; Λ(t, ·), 213 ; P, 190 ; 𝔇, 201 ; 𝔇(l_{2}), 201 ; ^{|}(0, τ]], 191 ; |g|_{l_{2}}^{2}, 190 ; ||g||_{n,p}, 190 ; || · ||_{1,2,p}, 187 ; || · ||_{n,p}, 187 ; || · ||_{p}, 187 ; || · ||_{H_{p}^{n}(τ)}, 191