Absolute continuity of solutions to reaction-diffusion equations with multiplicative noise

TL;DR

This paper proves that solutions to certain reaction-diffusion stochastic PDEs with multiplicative noise have absolutely continuous distributions at fixed points, using Malliavin calculus and well-posedness theory.

Contribution

It establishes absolute continuity of solutions' laws for a class of nonlinear stochastic PDEs driven by multiplicative noise, extending previous results to more general nonlinearities.

Findings

Solutions have absolutely continuous distributions at fixed points.

The proof employs Malliavin calculus and mild solution theory.

Results apply to reaction-diffusion equations on bounded domains.

Abstract

We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on $L^2(G)$, where $G$ is an open bounded domain in $\mathbb{R}^d$ with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

Equations

$$du(t)+Au(t)dt=f(u(t))dt+\sigma (u)BdW(t),u(0)=u_0,$$

$$du(t)+Au(t)dt=f(u(t))dt+B(u(t))dW(t),u(0)=u_0,$$

$$\displaystyle\big{\langle}f(x+y)-f(y),x^{*}\big{\rangle}$$

$$\displaystyle\big{\lVert}f(y)\big{\rVert}$$

$$\eta <\frac{1}{2}-\frac{1}{p},$$

$$E\underset{t\le T}{\sup }\Vert u(t){\Vert }_X^p\lesssim 1+E\Vert u_0{\Vert }_X^p.$$

$$\partial\lVert x\rVert=\bigl{\{}x^{*}\in X^{\prime}:\,\lVert x^{*}\rVert=1,\;\langle x^{*},x\rangle=1\bigr{\}}.$$

$$\big{\langle}-Ax+f(x+y),x^{*}\big{\rangle}\lesssim 1+\lVert y\rVert^{m}+\lVert x\rVert$$

$$\displaystyle\big{\langle}f(x+y),x^{*}\big{\rangle}$$

$$\displaystyle\lesssim 1+\lVert y\rVert^{m}+\big{\lvert}\big{\langle}f(y),x^{*}\big{\rangle}\big{\rvert}\lesssim 1+\lVert y\rVert^{m},$$

$$\frac{d}{2q}<\frac{1}{2}-\frac{1}{p},$$

$$\mathop{{}\mathbb{E}}\nolimits\sup_{t\leq T}\big{\lVert}u(t)\big{\rVert}_{C(\overline{G})}^{p}\lesssim 1+\mathop{{}\mathbb{E}}\nolimits\big{\lVert}u_{0}\big{\rVert}_{C(\overline{G})}^{p}.$$

$$\frac{d}{2q}<{\theta }^{\prime }<\theta <\frac{1}{2}-\frac{1}{p}.$$

$$E_{\theta}\hookrightarrow\bigl{(}L^{q},\operatorname{\mathsf{D}}(A)\bigr{)}_{\theta,\infty}\hookrightarrow\bigl{(}L^{q},\operatorname{\mathsf{D}}(A)\bigr{)}_{\theta^{\prime},1}\hookrightarrow\bigl{[}L^{q},\operatorname{\mathsf{D}}(A)\bigr{]}_{\theta^{\prime}},$$

$$\operatorname{\mathsf{D}}(A)=H_{q,D}^{2}(G):=\bigl{\{}\phi\in H_{q}^{2}(G):\;\phi|_{\partial G}=0\bigr{\}}.$$

$$\bigl{[}L^{q},H_{q,D}^{2}\bigr{]}_{\theta^{\prime}}=H_{q,D}^{2\theta^{\prime}}$$

$$\big{\langle}f(x+y)-\lambda(x+y)-(f(y)-\lambda y),x^{*}\big{\rangle}\leq 0,$$

$$\big{\langle}f(x+y)-f(y),x^{*}\big{\rangle}\leq\lambda\langle x,x^{*}\rangle\leq\lambda,$$

$$\big{\langle}f(x+y),x^{*}\big{\rangle}\leq\lambda+\big{\lvert}\big{\langle}f(y),x^{*}\big{\rangle}\big{\rvert}\lesssim\lambda+1+\lVert y\rVert^{m}.$$

$$\bigl{[}S(t)\phi\bigr{]}(x)=\int_{G}K_{t}(x,y)\phi(y)\,dy$$

$$E(W_h(t)W_g(s))=(t\wedge s)⟨Qh,g{⟩}_{L^2}$$

$$\int_{0}^{T}\!\!\int_{G}X(t,x)\,\mathcal{W}(dt,dx):=\sum_{k=1}^{\infty}\int_{0}^{T}\big{\langle}X(t,\cdot),\bar{e}^{k}\big{\rangle}_{L^{2}_{Q}}\,dW_{\bar{e}^{k}}(t),$$

$$\mathop{{}\mathbb{E}}\nolimits\bigg{\lvert}\int_{0}^{T}\!\!\int_{G}X(t,x)\,\mathcal{W}(dt,dx)\bigg{\rvert}^{2}=\mathop{{}\mathbb{E}}\nolimits\int_{0}^{T}\big{\lVert}X(t,\cdot)\big{\rVert}^{2}_{L^{2}_{Q}}\,dt.$$

$$\begin{array}{rl}u(t,x)& ={\int }_G{K}_t(x,y)u_0(y)dy+{\int }_0^t{\int }_G{K}_{t-s}(x,y)f(u(s,y))dyds\\ & +{\int }_0^t{\int }_G{K}_{t-s}(x,y)\sigma (u(s,y))\mathcal{W}(ds,dy).\end{array}$$

$$(s,y)\mapsto K_{t-s}(x,y)\sigma (u(s,y))$$

$${\int }_0^{t}S(t-s)\sigma (u(s))BdW(s)={\int }_0^t{\int }_G{K}_{t-s}(\cdot ,y)\sigma (u(s,y))\mathcal{W}(ds,dy)$$

$$\big{\lVert}K_{t-s}(x,\cdot)\sigma(u(s,\cdot))\big{\rVert}^{2}_{L^{2}_{Q}}=\sum_{k=1}^{\infty}\bigl{(}[S(t-s)\sigma(u(s))](\tilde{e}^{k})(x)\bigr{)}^{2},$$

$$\mathop{{}\mathbb{E}}\nolimits\int_{0}^{t}\sum_{k=1}^{\infty}\big{\lVert}[S(t-s)\sigma(u(s))](\tilde{e}^{k})\big{\rVert}^{2}_{L^{2}}ds<\infty,$$

$$\mathop{{}\mathbb{E}}\nolimits\int_{0}^{t}\big{\lVert}K_{t-s}(x,\cdot)\sigma(u(s,\cdot))\big{\rVert}^{2}_{L^{2}_{Q}}<\infty,$$

$$W(t)=\sum _{k=1}^{\infty }{e}^k{w}_k(t),$$

$${\int }_0^{t}S(t-s)\sigma (u(s))BdW(s)=\sum _{k=1}^{\infty }{\int }_0^t{\int }_G{K}_{t-s}(\cdot ,y)[\sigma (u(s))B{e}^k](y)dyd{w}_k(s).$$

Full text

2000 Mathematics Subject Classification: 60H07; 60H15.

Key words and phrases: Stochastic PDEs, reaction-diffusion equations, Malliavin calculus.

1 Introduction

Let $G$ be a bounded domain of $\mathbb{R}^{d}$, $d>1$, with smooth boundary. Consider a semilinear stochastic equation of the type

[TABLE]

where $A$ is the negative generator of an analytic semigroup on $L^{q}(G)$, $q\geq 2$, $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz continuous decreasing function with polynomial growth, $\sigma:\mathbb{R}\to\mathbb{R}$ is a Lipschitz continuous function, $B$ is a $\gamma$-Radonifying operator from $L^{2}(G)$ to $L^{q}(G)$, and $W$ is a cylindrical Wiener process on $L^{2}(G)$ (precise assumptions on the data of the problem are provided in §2 below). Then (1) admits a unique mild solution which is continuous in space and time. Our aim is to prove that the law of the random variable $u(t,x)$ is absolutely continuous with respect to Lebesgue measure for every fixed $(t,x)\in\mathbb{R}_{+}\times G$. It seems that, somewhat surprisingly, this natural question has not been addressed in the literature. In fact, all results of which we are aware about existence (and regularity) of the density of solutions to SPDEs with multiplicative noise deal with the case where $G$ is the whole space, $-A$ is the Laplacian, and the drift coefficient $f$ is (globally) Lipschitz continuous (see, e.g., [13, 15, 16, 19] and references therein). Our results do not rely on any one of these assumptions. In particular, we essentially just assume that the semigroup generated by $-A$ is self-adjoint and given by a family of kernel operators, so that, for instance, very large classes of elliptic second-order operators are allowed, and the function $f$ can be of polynomial type. Another major difference with respect to the above-mentioned works is that we rely almost exclusively on the interpretation of (1) as an equation for an $L^{q}(G)$-valued process, and that we view the pointwise Malliavin derivative of its solution as a process taking values in $L^{q}(G;H)$, where $H$ is a suitably chosen Hilbert space. This point of view, which allows us to rely on powerful techniques of the functional-analytic approach to stochastic evolution equations on UMD Banach spaces, is probably the most interesting aspect of this work. The more common random field interpretation of (1), that seems the only one used in previous work, at least in connection with techniques of the Malliavin calculus, is used here very sparingly, essentially only to take the pointwise Malliavin derivative of the solution to (1).

Existence and regularity of the density of solutions to semilinear heat equations with additive noise, i.e. for the easier case where $\sigma=1$ and $-A$ is the Laplacian, were obtained in [9]. Those results, however, depend heavily on the noise being additive, and cannot be extended to the general setting considered here. In fact, if the noise is additive, then the Malliavin derivative of the solution satisfies a deterministic equation with random coefficients, which yields quite strong estimates using pathwise arguments. On the other hand, if the noise is multiplicative, then the Malliavin derivative is only expected to satisfy a further stochastic evolution equation with quite singular initial condition, which is much more difficult to handle than the deterministic PDE arising in the case of additive noise. As a consequence, while in [9] we obtained existence as well as regularity of the density, here we can only show existence. As it is natural to expect, regularity could be obtained also in the case of multiplicative noise and Lipschitz continuous drift. However, we concentrate here only on the existence issue, and we shall deal with the regularity problem somewhere else, hopefully also in the general case where $f$ is monotone and polynomially bounded.

Let us briefly describe the main content of the paper. We first show existence and uniqueness of a unique mild solution $u$ to (1) which is continuous in space and time. This follows by relatively recent results on well-posedness in the mild sense for stochastic evolution equations in Banach spaces (see §2). Assuming that the semigroup generated by $-A$ is a family of kernel operators, the mild solution can be interpreted also in the sense of random fields. Considering first the case where $f$ is Lipschitz continuous, so that the mild solution is the unique fixed point of an operator $\Phi$, this reformulation allows to compute the Malliavin derivative of $\Phi$ applied to a class of sufficiently regular processes. Using estimates for stochastic convolutions in Banach spaces, we show that the fixed-point operator $\Phi$ leaves invariant a subspace of Malliavin differentiable processes with finite moment. This yields, by closability properties of the Malliavin derivative, that the unique mild solution to (1) is pointwise Malliavin differentiable. As a second step, we provide sufficient conditions ensuring that the Malliavin derivative is non-degenerate, adapting a method used in [16, theorem 5.2] for equations on $\mathbb{R}^{d}$ (see §3). This yields, as is well known, the pointwise absolute continuity of the law of the solution. As mentioned above, the results should be interesting in their own right, as equations in domains (in dimension higher than one) do not appear to have been considered in the literature. Finally, in the general case of equations of reaction-diffusion type, the pointwise absolute continuity of the law of the solution is treated by localization techniques, i.e. by means of the Bouleau-Hirsch criterion (see §4), and by convergence results for stochastic evolution equations with locally Lipschitz continuous coefficients in spaces of continuous functions.

Acknowledgments. The first-named author is sincerely grateful to Prof. S. Albeverio for several very pleasant stays at the Interdisziplinäres Zentrum für Komplexe Systeme, Universität Bonn, where most of the work for this paper was done. The second-named author is supported by the grant MTM2015-67802P.

2 Well-posedness in the space of continuous functions

We are going to establish well-posedness in the mild sense for the stochastic equation (1) in a space of continuous functions, using general well-posedness results for stochastic evolution equations in UMD Banach spaces (see [6, 21]). Assuming that the semigroup generated by $-A$ is a family of integral operators, we shall also show that the solution thus obtained can be viewed as a solution in the sense of random field (cf. [3, 22]).

2.1 Preliminaries

Let us consider the following stochastic evolution equation, posed on a general Banach space $X$:

[TABLE]

where $W$ is a cylindrical Wiener process on a Hilbert space $U$, and all other coefficients are specified below. The following well-posedness result is a slightly simplified version of [6, theorem 4.9].

Theorem 2.1**.**

Let $E$ be a UMD Banach space with type 2, such that $X$ is densely and continuously embedded in $E$ densely, and $A$ be a sectorial, accretive operator on $E$ such that the semigroup $S$ on $E$ generated by $-A$ restricts to a $C_{0}$-semigroup of contractions on $X$. Assume that $f:X\to X$ is locally Lipschitz continuous and there exists $m>0$ such that

[TABLE]

for all $x$, $y\in X$ and $x^{*}\in\partial\lVert x\rVert$. Let $p>2$ and assume that there exist numbers $\eta\in\mathbb{R}_{+}$, with

[TABLE]

such that $E_{\eta}:=\operatorname{\mathsf{D}}((I+A)^{\eta})$ is densely and continuously embedded in $X$. If $B:X\to\gamma(U,E)$ is locally Lipschitz continuous with linear growth, and $u_{0}\in L^{p}(\Omega;X)$, then there exists a unique $X$-valued mild solution to (2), which satisfies

[TABLE]

Here $\partial\lVert x\rVert$ stands for the subdifferential at $x$, in the sense of convex analysis, of the convex function $\lVert\cdot\rVert$, that is, denoting the dual of $X$ by $X^{\prime}$,

[TABLE]

Moreover, the notation $a\lesssim b$ means that there exists a constant $N$ such that $a\leq Nb$. To emphasize the dependence of $N$ on parameters $p_{1},\ldots,p_{n}$, we shall write $a\lesssim_{p_{1},\ldots,p_{n}}b$.

Remark 2.2*.*

In [6] the authors also require that

[TABLE]

for every $x\in\operatorname{\mathsf{D}}(A|_{X})$ and $x,y\in X$. Since we are assuming that $A$ is accretive in $X$, it follows that $\langle-Ax,x^{*}\rangle\leq 0$. Moreover,

[TABLE]

hence their condition, under our assumptions, is automatically satisfied.

Remark 2.3*.*

Further well-posedness results in $L^{q}$ spaces for semilinear parabolic SPDEs of accretive type, with more natural assumptions on the nonlinear drift term $f$, can be found in [7, 8, 10, 11, 12]. See also [2] for related results in spaces of continuous functions.

We shall also need some basic facts on interpolation. The real and the complex interpolation functors are denoted by $(\cdot,\cdot)$ and $[\cdot,\cdot]$, respectively. Moreover, we shall write $X\hookrightarrow Y$ to mean that $X$ is continuously embedded in $Y$.

Lemma 2.4**.**

Let $X$ and $Y$ be two Banach spaces forming an interpolation pair, $A$ a positive operator on $X$, and $\theta,\theta^{\prime}\in\mathopen{]}0,1\mathclose{[}$, $q,q^{\prime}\in[1,\infty]$ be constants. The following statements hold true:

• (a)

if $X\supset Y$, then $(X,Y)_{\theta,q}\hookrightarrow(X,Y)_{\theta^{\prime},q^{\prime}}$;

• (b)

$(X,Y)_{\theta,1}\hookrightarrow(X,Y)_{\theta,\infty}$;

• (c)

$(X,Y)_{\theta,1}\hookrightarrow[X,Y]_{\theta}\hookrightarrow(X,Y)_{\theta,\infty}$;

• (d)

\bigl{(}X,\operatorname{\mathsf{D}}(A)\bigr{)}_{\theta,1}\hookrightarrow\operatorname{\mathsf{D}}(A^{\theta})\hookrightarrow\bigl{(}X,\operatorname{\mathsf{D}}(A)\bigr{)}_{\theta,\infty}.

Proof.

All statements can be found in [20]. Specific references are provided for each result: (a) and (b) are parts of theorem 1.3.3, p. 25; (c) is a consequence of theorem 1, p. 64, taking into account definition 1.10.1, p. 61; (d) is part of theorem 1.15.2, p. 101. ∎

2.2 Existence of a unique mild solution

Let us now turn to equation (1), about which the following standing assumptions are assumed from now on.

Hypothesis 1**.**

(a) The operator $A$ is the realization on $L^{q}(G)$, $q\geq 2$, of a second-order strongly elliptic operator with $C^{\infty}$ coefficients, with Dirichlet boundary conditions. (b) The function $f:\mathbb{R}\to\mathbb{R}$ is an odd polynomial of degree $m>0$ with negative leading coefficient. (c) $W$ is a cylindrical Wiener process on $L^{2}(G)$ defined on a filtered probability space $(\Omega,\mathscr{F},(\mathscr{F}_{t})_{t\in[0,T]},\mathbb{P})$, with $T\in\mathbb{R}_{+}$, where $(\mathscr{F}_{t})_{t\in[0,T]}$ is the completion of the filtration generated by $W$.

It follows by (b) that $\lvert f(x)\rvert\lesssim 1+\lvert x\rvert^{m}$ for all $x\in\mathbb{R}$.

Proposition 2.5**.**

Assume that

[TABLE]

$\sigma\colon\mathbb{R}\to\mathbb{R}$* is locally Lipschitz continuous with linear growth, and $B\in\gamma(L^{2}(G),L^{q}(G))$. If $u_{0}\in L^{p}(\Omega;C(\overline{G}))$, then (1) admits a unique $C(\overline{G})$-valued mild solution $u$, which satisfies the estimate*

[TABLE]

Here $C(\overline{G})$ denotes the space of continuous functions on $\overline{G}$, the closure of $G$.

Proof.

We are going to verify that the assumptions of theorem 2.1 are satisfied. It follows from hypothesis (A) that, for any $q\geq 2$, $A$ is a sectorial, accretive operator on $L^{q}(G)$, and that the semigroup $S$ generated by $-A$ restricts to a $C_{0}$-semigroup on $C(\overline{G})$ (see, e.g., [17, theorem 3.5, pp. 213-214 and theorem 3.7, p. 217]). Moreover, denoting the evaluation operator on $C(\overline{G})$ associated to $f$ by the same symbol, it is not difficult to see that $f$ satisfies the assumptions of theorem 2.1 (detail can be found in [6, examples 4.2 and 4.5]). Moreover, one easily verifies that $u\mapsto\sigma(u)B$ is locally Lipschitz continuous and has linear growth as a map from $C(\overline{G})$ to $\gamma(U,L^{q}(G))$.

Let $\theta^{\prime}<\theta$ be such that

[TABLE]

Setting $E:=L^{q}:=L^{q}(G)$, let us show that $E_{\theta}\hookrightarrow C(\overline{G})$ densely: recall that, by lemma 2.4,

[TABLE]

where, by the characterization of $\operatorname{\mathsf{D}}(A)$ in [20, theorem 4.9.1, p. 334],

[TABLE]

Moreover, thanks to [20, theorem 3.3.4, p. 321], one has

[TABLE]

if $2\theta^{\prime}\neq 1/q$. Since $d>1$ and $2\theta^{\prime}>d$ by hypothesis, the latter condition is obviously satisfied, hence $E_{\theta}\hookrightarrow H_{q,D}^{2\theta^{\prime}}\subset H_{q}^{2\theta^{\prime}}$. Finally, the Sobolev embedding theorem (cf. [20, theorem 4.6.1, p. 328]) yields $H_{q}^{2\theta^{\prime}}\hookrightarrow C(\overline{G})$, assuming that $2\theta^{\prime}>d/q$, which is satisfied by hypothesis. We have thus shown that all assumptions of theorem 2.1 are met, hence the claim is proved. ∎

Note that $p>2$ imply that, for $q$ large enough, the hypothesis $d/(2q)<1/2-1/p$ is always satisfied.

Remark 2.6*.*

Instead of assuming that $f$ is an odd polynomial with negative leading coefficient, one could also assume that $f:\mathbb{R}\to\mathbb{R}$ is locally Lipschitz continuous, polynomially bounded, and quasi-monotone, i.e. that there exists $\lambda>0$ such that $x\mapsto\lambda x-f(x)$ is increasing. In fact, assume that there exists $m>0$ such that $\lvert f(x)\rvert\lesssim 1+\lvert x\rvert^{m}$. By dissipativity of $f-\lambda I$,

[TABLE]

hence

[TABLE]

and

[TABLE]

2.3 Mild solution as random field

We assume from now on, in addition to hypothesis 1, the following condition on the semigroup $S$ generated by $-A$.

Hypothesis 2**.**

The semigroup $S=(S(t))_{t\geq 0}$ is sub-Markovian (i.e. $S(t)$ is positive and contracting in $L^{\infty}(G)$ for all $t\geq 0$) and admits a kernel, in the sense that there exists a function $K:\mathbb{R}_{+}\times G^{2}\to\mathbb{R}_{+}$ such that

[TABLE]

for every $\phi\in L^{q}(G)$, $q\geq 1$.

Let $Q:=BB^{*}$, which is a symmetric and non-negative definite bounded operator. Recall that a cylindrical $Q$-Wiener process on $L^{2}:=L^{2}(G)$ is a Gaussian family of random variables $\mathcal{W}:=\{W_{h}(t),\,h\in L^{2},t\geq 0\}$ such that, for all $s,t\geq 0$ and $h,g\in L^{2}$, $\mathop{{}\mathbb{E}}\nolimits(W_{h}(t))=0$ and

[TABLE]

(in spite of the slight abuse of notation, no confusion should arise with the cylindrical Wiener process $W$). Let $L^{2}_{Q}$ be the Hilbert space defined as the completion of $L^{2}$ with respect to the scalar product $\langle h,g\rangle_{L^{2}_{Q}}:=\langle Qh,g\rangle_{L^{2}}$. Note that, denoting the pseudoinverse of $Q^{1/2}$ by $Q^{-1/2}$, if $(e^{k})_{k\in\mathbb{N}}$ is a basis of $L^{2}$, then $(\bar{e}^{k}):=(Q^{-1/2}e^{k})$ is a basis of $L^{2}_{Q}$ . One can define stochastic integrals with respect to $\mathcal{W}$ as follows (see, e.g., [4, Sec. 2]): let $\{X(t,x):\,(t,x)\in[0,T]\times G\}$ be a predictable process in $L^{2}(\Omega\times[0,T];L^{2}_{Q})$. Then

[TABLE]

and the isometry property reads

[TABLE]

In order to prove that the Malliavin derivative of the solution $u$ of (1) satisfies a stochastic equation, we need to verify that $u$ can be interpreted as a mild solution to (1) in the sense of random fields (see, e.g., [3, 4, 22]). This is indeed the case (cf. the analogous result for equations with additive noise in [9]).

Proposition 2.7**.**

Let the assumptions of proposition 2.5 be satisfied. For any $(t,x)\in[0,T]\times G$, set $u(t,x):=[u(t)](x)$, where $u$ is the unique $C(G)$-valued mild solution to (1). Then for any $(t,x)\in\mathopen{]}0,T]\times G$,

[TABLE]

Proof.

As in the proof of [9, proposition 3.1], it suffices to show that, for every $t\in\mathopen{]}0,T]$ and for almost every $x\in G$, the process

[TABLE]

belongs to $L^{2}(\Omega\times[0,T];L^{2}_{Q})$ and that

[TABLE]

as an equality in $L^{2}$. Recalling that $(\bar{e}^{k})=(Q^{-1/2}e^{k})$, is a basis of the Hilbert space $L^{2}_{Q}$, one easily verifies that

[TABLE]

where $(\tilde{e}^{k}):=(Q^{1/2}e^{k})$ is a basis of $Q^{1/2}(L^{2})$. Note that

[TABLE]

because the stochastic integral on the left-hand side of (4) is well defined. Thus, for almost all $x\in G$,

[TABLE]

so the stochastic integral on the right-hand side of (4) is well defined. It remains equality in (4). Using the standard formal expansion of the cylindrical Wiener process $W$ as

[TABLE]

where $w_{k}:=W_{\bar{e}^{k}}$, $k\geq 1$, form a family of independent standard one-dimensional Wiener processes, one has

[TABLE]

Then (4) follows taking into account the definition (3) and that $BB^{*}=Q$. ∎

3 Equations with Lipschitz continuous coefficients

We assume throughout this section that the coefficients $f$ and $\sigma$ in equation (1) are Lipschitz continuous. We are going to prove that, for any fixed $(t,x)\in(0,T]\times G$, the law of the solution $u(t,x)$ to (1) is absolutely continuous with respect to the Lebesgue measure. For this, note that the Gaussian space in which we will make use of the Malliavin calculus’ techniques is determined by the isonormal Gaussian process on the Hilbert space $H:=L^{2}(0,T;L^{2}_{Q})$ which can be naturally associated to the cylindrical $Q$-Wiener process $\mathcal{W}$ defined in the previous section (see [14]).

We will first deal with the Malliavin differentiability of the solution, and then we shall provide sufficient conditions implying that the pointwise Malliavin derivative is non-degenerate.

We need further assumptions, that will be assumed to hold from now on.

Hypothesis 3**.**

One has

[TABLE]

Moreover, $B\in\gamma(L^{2},L^{q})$ and $u_{0}\in C(\overline{G})$.

Hypothesis 4**.**

The semigroup $S$ is self-adjoint and Markovian.

Recall also that we assume that hypotheses 1 and 2 are in force throughout. By proposition 2.5, it follows that (1) admits a unique $C(\overline{G})$-valued mild solution $u$, and that (1) can also be written as an equality of random fields.

3.1 Pointwise Malliavin differentiability of the solution

The main result of this section is the following.

Theorem 3.1**.**

Let $u\in L^{p}(\Omega;C([0,T];C(\overline{G})))$ be the unique mild solution to (1). Then

[TABLE]

and the family of Malliavin derivatives $\{Du(t,x)\}_{(t,x)\in[0,T]\times G}$ satisfies the following linear equation in $H$:

[TABLE]

where

[TABLE]

and $F$, $\Sigma\colon\Omega\times[0,T]\times G\to\mathbb{R}$ are adapted bounded random fields.

The stochastic integral in (5) must be interpreted as an $H$-valued integral with respect to the cylindrical $Q$-Wiener process $\mathcal{W}$ (see, e.g., [16, §3]).

The following estimate plays an important role in the proof theorem 3.1 as well as in several further developments. We shall write $E^{q}_{\eta}$, for any $q\geq 1$ and $\eta>0$, to denote $(I+A)^{-\eta}L^{q}$.

Lemma 3.2**.**

Let $v\in L^{p}(\Omega;C([0,T];C(\overline{G})))$ be adapted and $w:\Omega\times[0,T]\times G\to H$ be the process defined as

[TABLE]

For any $\eta\in\mathopen{]}d/(2q),1/2-1/p\mathclose{[}$ one has

[TABLE]

Proof.

Since $H=L^{2}(0,T;L^{2}_{Q})$ and $\langle{Qh},{h}\rangle=\lVert B^{*}h\rVert^{2}_{L^{2}}$ for every $h\in L^{2}_{Q}$, denoting a complete orthonormal basis of $L^{2}$ by $(e^{k})_{k\geq 1}$, it follows by Plancherel’s theorem that

[TABLE]

where we have used the integral representation of the semigroup $S$ in the last step. Let $(\gamma_{k})$ be a sequence of independent standard Gaussian random variable on an auxiliary probability space $\Omega^{\prime}$. Then

[TABLE]

hence also, by Minkowski’s inequality and the embedding $E^{q}_{\eta}\hookrightarrow L^{\infty}$,

[TABLE]

The proof of theorem 3.1 uses a maximal inequality for stochastic convolutions, that is a special (simpler) case of [21, proposition 4.2]. We shall use the notation $R\diamond F$ to denote the process

[TABLE]

where $R$ is an analytic semigroup of contractions on a UMD Banach space $E$ and $F\colon\Omega\times\mathbb{R}_{+}\to\mathscr{L}(L^{2},E)$ is an $L^{2}$-strongly measurable and adapted process. Denoting the generator of $R$ by $-C$, we shall write $E_{\eta}$, for any $\eta>0$, to denote $\operatorname{\mathsf{D}}((I+C)^{\eta})$.

Proposition 3.3**.**

Let $\alpha\in\mathopen{]}0,1/2\mathclose{[}$, $p>2$, $\theta\geq 0$ be such that

[TABLE]

and $T>0$. There exists $\varepsilon>0$ such that

[TABLE]

We shall also need a deep result by Pisier (see [18, theorem 1.2 and remark 1.8] as well as [23, p. 5730]) on vector-valued extensions of analytic semigroup, according to which hypothesis 4 implies that $(S(t)\otimes I_{H})_{t\geq 0}$, where $I$ denotes the identity of $H$, admits a (unique) extensions from $L^{q}\otimes H$ to $L^{q}(H)$, denoted by $S_{H}$, which is again analytic. Let $A_{H}$ denote the negative generator of $S_{H}$ and $(\lambda+A_{H})_{\lambda>0}^{-1}$ its resolvent. The Laplace transform identity

[TABLE]

implies that $(\lambda+A_{H})^{-1}$ coincides with the unique continuous linear extension of $(\lambda+A)^{-1}\otimes I_{H}$ to $L^{q}(H)$. By hypothesis 3 there exists $\eta\in\mathopen{]}d/(2q),1/2-1/p\mathclose{[}$ such that $\operatorname{\mathsf{D}}(A^{\eta})\hookrightarrow L^{\infty}$, hence $(I+A)^{-\eta}\in\mathscr{L}(L^{q},L^{\infty})$. Since $(I+A)^{-\eta}$ is positivity preserving by hypothesis 2, $(I+A)^{-\eta}$ admits a unique extension to a continuous linear operator from $L^{q}(H):=L^{q}(G;H)$ to $L^{\infty}(H):=L^{\infty}(G;H)$, with the same norm (see, e.g., [5, theorem 12.2]). By the above, recalling well-known expressions for fractional powers of closed operators (see, e.g., [17, §2.6]), this extension coincides with $(I+A_{H})^{-\eta}$. Therefore, setting $E^{q}_{\eta}(H):=(I+A)^{-\eta}L^{q}(H)$, we have $E^{q}_{\eta}(H)\hookrightarrow L^{\infty}(H)$.

Proof of theorem 3.1.

Let $\Phi$ be the fixed-point operator associated to equation (1), i.e.

[TABLE]

It follows by the (the proof) of theorem 2.1 that the operator $\Phi$, or a suitable power of it, is a contractive endomorphism of $L^{p}(\Omega;C([0,T];C(\overline{G})))$. We are going to show that, for any $p>2$, there exists $T_{0}>0$, a positive constants $c<1$ depending on $T_{0}$, and a positive constant $N$ depending on the $L^{p}(\Omega;C([0,T];C(\overline{G})))$ norm of $v$, such that

[TABLE]

Let $v\in L^{p}(\Omega;C([0,T];C(\overline{G})))$ be such that $Dv\in L^{\infty}(0,T;L^{p}(\Omega;L^{\infty}(G;H)))$. Writing

[TABLE]

well-known criteria of Malliavin calculus imply that the Malliavin derivatives of all terms on the right-hand side exist, so that D\bigl{[}\Phi(v)\bigr{]}(t,x) can be written as the right-hand side of (5) with $u$ replaced by $v$. The proof of (6) will be split in several steps, where each term appearing in the expression of $D\Phi(v)$ is estimated.

Step 1. Let us set, for every $(t,x)$, $(\tau,z)\in[0,T]\times G$,

[TABLE]

Let $\eta\in\mathopen{]}d/(2q),1/2-1/p\mathclose{[}$. Lemma 3.2 yields

[TABLE]

where

[TABLE]

This implies

[TABLE]

where the last term on the right-hand side is finite by assumption.

Step 2. Let $\alpha<1/2$ such that $\eta<\alpha-1/p$. Recalling that $E^{q}_{\eta}(H)\hookrightarrow L^{\infty}(H)$, Minkowski’s and Jensen’s inequality yield

[TABLE]

Since $\eta<\alpha-1/p$ by assumption, we have $-2\eta>-2\alpha+2/p$, hence $-2\eta=-2\alpha+2/p+\varepsilon$, with $\varepsilon>0$. Then

[TABLE]

As the measure $\mu$ on $[0,t]$ defined as

[TABLE]

is a probability measure, it follows by Jensen’s inequality that

[TABLE]

Therefore

[TABLE]

Step 3. Using again the continuous embedding $E^{q}_{\eta}(H)\hookrightarrow L^{\infty}(H)$, we have

[TABLE]

where the third inequality follows by proposition 3.3, as $L^{q}(H)$ is a UMD Banach space and $\eta<\alpha-1/p$, and the fourth estimate follows by Fubini’s theorem and the embedding

[TABLE]

which holds because $L^{q}(H)$ has type 2. Since $Dv(s)\in L^{\infty}(H)$ by assumption and $\Sigma\in L^{\infty}([0,T]\times G)$ by the Lipschitz continuity of $\sigma$, it follows that

[TABLE]

hence

[TABLE]

Proceeding as in the previous step, we obtain

[TABLE]

therefore, by Tonelli’s theorem,

[TABLE]

where

[TABLE]

hence

[TABLE]

Step 4. Setting

[TABLE]

the estimates in the previous steps can be written as

[TABLE]

hence, using the notation $h^{*}(s):=\sup_{r\leq s}\lvert h(r)\rvert$ for any function $h\colon\mathbb{R}\to\mathbb{R}$ for which it makes sense,

[TABLE]

thus also

[TABLE]

from which (6) follows.

Let $u_{0}$ be identified with the process equal to $u_{0}$ for all $t\in[0,T]$, which clearly belongs to $L^{p}(\Omega;C([0,T];C(\overline{G})))$ and is such that $Du_{0}\in L^{\infty}(0,T;L^{p}(\Omega;L^{\infty}(G;H)))$, and introduce the sequence of processes $(u_{n})$, $u_{n}:=\Phi(u_{n-1})$. Then $u_{n}$ converges to $u$ in $L^{p}(\Omega;C([0,T];C(\overline{G})))$, possibly along a subsequence of the type $(kn)$, with constant $k$ (if $\Phi$ is not a contraction, but $\Phi^{k}$ is). In particular, $(u_{n})$ is bounded in $L^{p}(\Omega;C([0,T];C(\overline{G})))$. This in turn implies, thanks to (6), that $(Du_{n})$ is bounded in $L^{\infty}([0,T_{0}]\times G;L^{p}(\Omega;H))$. Let us show that this actually implies that $(Du_{n})$ is bounded in $L^{\infty}([0,T]\times G;L^{p}(\Omega;H))$. In fact, setting

[TABLE]

we have already shown that

[TABLE]

and that $(\phi^{*}_{n}(T_{0}))_{n}$ is bounded. We now proceed by induction: assuming that $(\phi^{*}_{n}(jT_{0}))_{n}$ is bounded, let us show that $(\phi^{*}_{n}((j+1)T_{0}))_{n}$ is also bounded. Let $jT_{0}<t\leq(j+1)T_{0}$. We have

[TABLE]

where $t>jT_{0}$ implies $t-s>jT_{0}-s$ and $(t-s)^{-2\alpha}<(jT_{0}-s)^{-2\alpha}$, hence

[TABLE]

so that

[TABLE]

where

[TABLE]

This in turn implies, taking the supremum over $[0,(j+1)T_{0}]$,

[TABLE]

Since $\phi^{*}_{n}(jT_{0})$ is bounded uniformly with respect to $n$ by the inductive assumption, we deduce that $\phi_{n}^{*}((j+1)T_{0})$ is bounded uniformly over $n$ as well, thus completing the inductive argument. This implies, by a standard argument based on the closure of the Malliavin derivative, that $u\in L^{\infty}([0,T]\times G;\mathbb{D}^{1,p})$.

Finally, the equation for $Du$ follows immediately by differentiating equation (2.7). ∎

3.2 Non-degeneracy of the Malliavin derivative

This section is devoted to study, for any fixed $(t,x)\in\mathopen{]}0,T]\times G$, the norm of the Malliavin derivative of $u(t,x)$. Together with the results of the previous section, we will deduce the existence of the density for the law of the random variable $u(t,x)$. Recall that throughout the section we are assuming that $f$ and $\sigma$ are globally Lipschitz continuous functions.

We will need an estimate for the norm of $Du(t,x)$ in

[TABLE]

Proposition 3.4**.**

Let $0\leq a<b\leq T$, $p>2$, and $\eta\in\mathopen{]}d/(2q),1/2-1/p\mathclose{[}$. There exists a positive constant $N$, independent of $a$ and $b$, such that

[TABLE]

Proof.

Repeating the proof of theorem 3.1 with $H$ replaced by $H(a,b)$, we get

[TABLE]

and, by lemma 3.2,

[TABLE]

where

[TABLE]

Therefore

[TABLE]

In the next result, we establish sufficient conditions under which the norm of the Malliavin derivative of $u(t,x)$ does not vanish, almost surely.

Proposition 3.5**.**

Assume that there exists a constant $c>0$ such that $\lvert\sigma(z)\rvert\geq c$ for all $z\in\mathbb{R}$ and that $Q$ is positivity preserving. Let $(t,x)\in\mathopen{]}0,T]\times G$, $\alpha\in\mathopen{]}0,1/2\mathclose{[}$, and $\eta\in\mathopen{]}d/(2q),\alpha-1/p\mathclose{[}$. If there exist $\beta\in\mathopen{]}0,1-\alpha-\eta]$ such that

[TABLE]

then $\lVert Du(t,x)\rVert_{H}>0$ almost surely.

Proof.

We are going to estimate $\mathbb{P}(\lVert Du(t,x)\rVert_{H}\leq 1/n)$ for $n\in\mathbb{N}$ and pass to the limit as $n\to\infty$. Let $\delta\in\mathopen{]}0,1\mathclose{[}$, and set, for compactness of notation, $H_{\delta}:=H(t-\delta,t)$. The obvious inequality $\lVert a+b\rVert\geq\lVert a\rVert-\lVert b\rVert$ applied to the expression of $Du$ given by theorem 3.1 yields

[TABLE]

Hence, simplifying the notation a bit and denoting the second term within the norm on the right-hand side by $Y$,

[TABLE]

Since $Q$ as well as the semigroup $S$ is positivity preserving, hence $K$ is positive, and $\sigma:\mathbb{R}\to\mathbb{R}$ is continuous, we have

[TABLE]

This implies that we can use Chebyshev’s inequality to write, for $n$ sufficiently large,

[TABLE]

where, thanks to theorem 3.1 and proposition 3.4,

[TABLE]

Taking the limit as $n\to\infty$, we are left with

[TABLE]

Since this inequality holds for every $\delta\in\mathopen{]}0,1\mathclose{[}$, and the limit of the right-hand side as $\delta\to 0$ is zero by assumption, it follows that \mathbb{P}\bigl{(}\lVert Du(t,x)\rVert_{H}=0\bigr{)}=0. ∎

As an immediate consequence of the above result and of theorem 3.1 we obtain sufficient conditions for the pointwise absolute continuity of the law of the mild solution to (1), thanks to well-known criteria of the Malliavin calculus (see, e.g., [14, theorem 2.1.3]).

Theorem 3.6**.**

Let $u\in L^{p}(\Omega;C([0,T];C(\overline{G})))$ be the unique mild solution to equation (1), with $f$ and $\sigma$ Lipschitz continuous and $u_{0}\in C(\overline{G})$. Assume that there exists $c>0$ such that $\lvert\sigma(z)\rvert\geq c>0$ for all $z\in\mathbb{R}$ and $Q=BB^{*}$ is positivity preserving. Let $(t,x)\in\mathopen{]}0,T]\times G$, $\alpha\in\mathopen{]}0,1/2\mathclose{[}$, and $\eta\in\mathopen{]}d/(2q),\alpha-1/p\mathclose{[}$. If there exist $\beta\in\mathopen{]}0,1-\alpha-\eta]$ such that (7) is fulfilled, then the law of the random variable $u(t,x)$ is absolutely continuous with respect to Lebesgue measure.

Example 3.7*.*

Assume that $A$ has compact resolvent in $L^{2}$. Since $A$ is accretive and self-adjoint, there exist an orthonormal basis $(e^{k})_{k\in\mathbb{N}}$ of $L^{2}$ and a sequence $(\lambda_{k})_{k\in\mathbb{N}}\geq 0$ such that $e^{k}\in\operatorname{\mathsf{D}}(A)$, $Ae^{k}=\lambda_{k}e^{k}$ and $\lim_{k\to\infty}\lambda_{k}=+\infty$. Moreover, let $B=(I+A)^{-m}$, with $m\in\mathbb{N}$, and fix $(t,x)\in\mathopen{]}0,T]\times G$. Since $Q=(I+A)^{-2m}$, one has, for any $\delta\in\mathopen{]}0,1\mathclose{[}$,

[TABLE]

Moreover, we have that

[TABLE]

Hence

[TABLE]

Assuming that $x\in G$ is such that there exists $k\in\mathbb{N}$ for which $e^{k}(x)\neq 0$, the quantity

[TABLE]

is strictly positive. Therefore we have $\lVert K_{\cdot}(x,\cdot)\rVert^{2}_{H(0,\delta)}\geq C_{x}\delta$, i.e.

[TABLE]

which implies that condition (7), hence also the assumptions of theorem 3.6, are satisfied if we can find $\alpha$ and $\eta$ such that $1-\alpha-\eta>1/2$. This is possible if $m$ is sufficiently large, so that $B\in\gamma(L^{2},L^{q})$ with $q$ large and $d/(2q)$ is smaller than, say, $1/4$.

4 Reaction-diffusion equations

Let us now consider equation (1) in the general case, i.e. assuming that $f\colon\mathbb{R}\to\mathbb{R}$ is an odd polynomial with negative leading coefficient. As already observed, we could also assume that $x\mapsto f(x)-\lambda x$ is decreasing for some $\lambda\geq 0$, locally Lipschitz continuous, and with polynomial growth.

Let $u_{0}\in C(\overline{G})$, and $u\in L^{p}(\Omega;C([0,T];C(\overline{G})))$ be the unique mild solution to (1), the existence of which is guaranteed by proposition 2.5. For every $n\in\mathbb{N}$, consider the function $f_{n}\colon\mathbb{R}\to\mathbb{R}$ defined as

[TABLE]

Then $f_{n}$ is Lipschitz continuous, and the equation

[TABLE]

admits a unique mild solution $u_{n}\in L^{p}(\Omega;C([0,T];C(\overline{G})))$. Moreover, by construction of $u$ (see [6]), $u_{n}$ coincides with $u$ on the stochastic interval $[\![0,T_{n}]\!]$, where the stopping time $T_{n}$ is defined as

[TABLE]

and $\lim_{n\to\infty}T_{n}=T$ almost surely. In particular, $u_{n}\to u$ in $L^{r}(\Omega;C([0,T];C(\overline{G})))$ for all $r\in[1,p\mathclose{[}$. Let $t\in\mathopen{]}0,T]$ be arbitrary but fixed and set, for every $n\in\mathbb{N}$,

[TABLE]

Since $(T_{n})$ is a sequence of stopping times monotonically increasing to $T$ as $n\to\infty$, $(\Omega_{n})$ is a sequence in $\mathscr{F}$ monotonically increasing to $\Omega$ as $n\to\infty$. Clearly $\{t\}\times\Omega_{n}\subset[\![0,T_{n}]\!]$, hence $u(t)=u_{n}(t)$ on $\Omega_{n}$, as an identity in $C(\overline{G})$. This implies that $u(t,x)=u_{n}(t,x)$ on $\Omega_{n}$ for every $x\in G$. Moreover, as $f_{n}$ is Lipschitz continuous, theorem 3.1 implies that $u_{n}(t,x)\in\mathbb{D}^{1,p}$ for every $x\in G$, for all $p\geq 1$. We have thus shown that $u(t,x)\in\mathbb{D}^{1,p}_{\mathrm{loc}}$, with localizing sequence $(\Omega_{n},u_{n}(t,x))$ (cf. [1, §III] or [14, §1.3.5]). This implies that $u(t,x)$ is Malliavin differentiable, i.e. that there exists a random variable $Du(t,x)$, independent of the chosen localizing sequence, such that $Du(t,x)=Du_{n}(t,x)$ on $\Omega_{n}$.

We are now in the position to state and prove the main result of the paper.

Theorem 4.1**.**

Let $u\in L^{p}(\Omega;C([0,T];C(\overline{G})))$ be the unique mild solution to equation (1) with initial datum $u_{0}\in C(\overline{G})$. Assume that $Q=BB^{*}$ is positivity preserving and that there exists $c>0$ such that $\lvert\sigma(z)\rvert\geq c>0$ for all $z\in\mathbb{R}$. Let $(t,x)\in\mathopen{]}0,T]\times G$, $\alpha\in\mathopen{]}0,1/2\mathclose{[}$, and $\eta\in\mathopen{]}d/(2q),\alpha-1/p\mathclose{[}$. If there exist $\beta\in\mathopen{]}0,1-\alpha-\eta]$ such that

[TABLE]

then the law of the random variable $u(t,x)$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.

Proof.

Let $(t,x)\in G_{T}$ be arbitrary but fixed. Then, by the Bouleau-Hirsch’ criterion (see [1, proposition 7.1.4]), it suffices to prove that $\lVert Du(t,x)\rVert_{H}>0$ almost surely. Since $f_{n}$ is Lipschitz continuous for all $n\in\mathbb{N}$, $\lVert Du(t,x)\rVert_{H}>0$ on $\Omega_{n}$ for all $n\in\mathbb{N}$. This readily implies that $\lVert Du(t,x)\rVert_{H}>0$ almost surely: assume by contradiction that there exists $\Omega^{\prime}\subset\Omega$ with strictly positive probability such that $\lVert Du(t,x)\rVert_{H}=0$ on $\Omega^{\prime}$. Since $\Omega_{n}$ increases monotonically to $\Omega$, there exists $n_{0}\in\mathbb{N}$ such that $\mathbb{P}\left(\Omega^{\prime\prime}\right)>0$, where $\Omega^{\prime\prime}:=\Omega_{n_{0}}\cap\Omega^{\prime}$. In particular, by definition of $\Omega_{n_{0}}$, one has $\lVert Du(t,x)\rVert_{H}>0$ on $\Omega^{\prime\prime}$ because $\Omega^{\prime\prime}\subset\Omega_{n_{0}}$. This is clearly a contradiction, because $\Omega^{\prime\prime}\subset\Omega^{\prime}$. The claim is thus proved. ∎

Remark 4.2*.*

Very minor adjustments allow to consider the case where $\sigma\colon\mathbb{R}\to\mathbb{R}$ is locally Lipschitz continuous with linear growth. In fact, the construction of a unique global solution is obtained again by récollement of local solutions (see [6]), and the above reasoning can be repeated almost verbatim.

Remark 4.3*.*

The setting of example 3.7 obviously satisfies the assumptions of theorem 4.1.