On the well-posedness of SPDEs with singular drift in divergence form

TL;DR

This paper establishes the existence and uniqueness of strong solutions for a class of stochastic PDEs with singular divergence-form drift and multiplicative noise, broadening the understanding of well-posedness in such equations.

Contribution

It proves well-posedness for SPDEs with maximal monotone divergence drift, extending previous results to more general multivalued cases with minimal assumptions.

Findings

Existence of strong solutions under broad conditions

Uniqueness of solutions for the class of SPDEs considered

Self-contained proof of well-posedness

Abstract

We prove existence and uniqueness of strong solutions for a class of second-order stochastic PDEs with multiplicative Wiener noise and drift of the form $\operatorname{div} \gamma(\nabla \cdot)$, where $\gamma$ is a maximal monotone graph in $\mathbb{R}^n \times \mathbb{R}^n$ obtained as the subdifferential of a convex function satisfying very mild assumptions on its behavior at infinity. The well-posedness result complements the corresponding one in our recent work arXiv:1612.08260 where, under the additional assumption that $\gamma$ is single-valued, a solution with better integrability and regularity properties is constructed. The proof given here, however, is self-contained.

Equations

$$du(t)-\mathrm{div}\gamma (\nabla u(t))dt\ni B(t,u(t))dW(t),u(0)=u_0,$$

$$\underset{|x|\to \infty }{\lim }\frac{k(x)}{|x|}=+\infty ,\underset{|x|\to \infty }{\mathrm{limsup}}\frac{k(-x)}{k(x)}<+\infty$$

$$u\in L^1(0,T;W_0^{1,1}(D))\text{and}B(\cdot ,u)\in L^2(0,T;{\mathcal{L}}^2(U,L^2(D)))\mathbb{P}\text{-a.s.};$$

$$\eta \in L^1(0,T;L^1(D{)}^n)\mathbb{P}\text{-a.s.},\eta \in \gamma (\nabla u)\text{a.e. in }\Omega \times (0,T)\times D;$$

$$u(t)-{\int }_0^t\mathrm{div}\eta (s)ds=u_0+{\int }_0^{t}B(s,u(s))dW(s)\mathbb{P}\text{-a.s.}\forall t\in [0,T].$$

$$\underset{t\le T}{\sup }E\Vert u(t){\Vert }_{L^2(D)}^2+E{\int }_0^T\Vert u(t){\Vert }_{W_0^{1,1}(D)}dt<\infty ,$$

$$E{\int }_0^T\Vert \eta (t){\Vert }_{L^1(D{)}^n}dt<\infty ,$$

$$\displaystyle\mathop{{}\mathbb{E}}\int_{0}^{T}\bigl{(}\lVert{k(\nabla u(t))}\rVert_{L^{1}(D)}+\lVert{k^{*}(\eta(t))}\rVert_{L^{1}(D)}\bigr{)}\,dt<\infty.$$

$$du(t)-\mathrm{div}\gamma (\nabla u(t))dt\ni G(t)dW(t),u(0)=u_0,$$

$$d{u}_{\lambda }(t)-\mathrm{div}{\gamma }_{\lambda }(\nabla u_{\lambda }(t))dt-\lambda \Delta u_{\lambda }(t)dt=G(t)dW(t),u_{\lambda }(0)=u_0,$$

$$E\Vert u_{\lambda }{\Vert }_{C([0,T];L^2(D))}^2+E{\int }_0^T\Vert u_{\lambda }(t){\Vert }_{H_0^1(D)}^{2}dt<\infty$$

$$u_{\lambda }(t)-{\int }_0^t\mathrm{div}{\gamma }_{\lambda }(\nabla u_{\lambda }(s))ds-\lambda {\int }_0^t\Delta u_{\lambda }(s)ds=u_0+{\int }_0^{t}G(s)dW(s)$$

$$\Vert u_{\lambda }{\Vert }_{L_{\omega }^2{C}_t{L}_x^2}+{\lambda }^{1/2}\Vert \nabla u_{\lambda }{\Vert }_{L_{t,\omega ,x}^2}+\Vert {\gamma }_{\lambda }(\nabla u_{\lambda })\cdot \nabla u_{\lambda }{\Vert }_{L_{t,\omega ,x,}^1}$$

$$\displaystyle\hskip 30.00005pt<N\bigl{(}\lVert{u_{0}}\rVert_{L^{2}_{\omega,x}}+\lVert{G}\rVert_{L^{2}_{t,\omega}\mathscr{L}^{2}(H,L^{2}_{x})}\bigr{)}.$$

$$\Vert u_{\lambda }(t){\Vert }^2+2{\int }_0^t{\int }_D\gamma (\nabla u_{\lambda }(s))\cdot \nabla u_{\lambda }(s)dxds+2\lambda {\int }_0^t\Vert \nabla u_{\lambda }(s){\Vert }^{2}ds$$

$$=\Vert u_0{\Vert }^2+2{\int }_0^t{u}_{\lambda }(s)G(s)dW(s)+{\int }_0^t\Vert G(s){\Vert }_{{\mathcal{L}}^2(H,L_x^2)}^{2}ds,$$

$$E\Vert u_{\lambda }{\Vert }_{C_t{L}_x^2}^2+E{\int }_0^T{\int }_D{\gamma }_{\lambda }(\nabla u_{\lambda }(s))\cdot \nabla u_{\lambda }(s)dxds+\lambda E\Vert \nabla u_{\lambda }{\Vert }_{L_{t,x}^2}^2$$

$$\lesssim E\Vert u_0{\Vert }^2+E\Vert G{\Vert }_{L_t^2{\mathcal{L}}^2(H,L_x^2)}^2+E\underset{t\in [0,T]}{\sup }\left|{\int }_0^t{u}_{\lambda }(s)G(s)dW(s)\right|,$$

$$E\underset{t\in [0,T]}{\sup }\left|{\int }_0^t{u}_{\lambda }(s)G(s)dW(s)\right|$$

$$\le \epsilon E\Vert u_{\lambda }{\Vert }_{C_t{L}_x^2}^2+N(\epsilon )E{\int }_0^T\Vert G(s){\Vert }_{{\mathcal{L}}^2(H,L_x^2)}^{2}ds$$

$$\gamma_{\lambda}(x)\in\partial k\bigl{(}(I+\lambda\gamma)^{-1}x\bigr{)}=\gamma\bigl{(}(I+\lambda\gamma)^{-1}x\bigr{)}\qquad\forall x\in\mathbb{R}^{n},$$

$$\displaystyle k\bigl{(}(I+\lambda\gamma)^{-1}x\bigr{)}+k^{*}\bigl{(}\gamma_{\lambda}(x)\bigr{)}$$

$$={\gamma }_{\lambda }(x)\cdot x-\lambda {\left|{\gamma }_{\lambda }(x)\right|}^2\le {\gamma }_{\lambda }(x)\cdot x\forall x\in {\mathbb{R}}^n.$$

$$\mathop{{}\mathbb{E}}\int_{0}^{T}\!\!\int_{D}k^{*}\bigl{(}\gamma_{\lambda}(\nabla u_{\lambda})\bigr{)}\leq\mathop{{}\mathbb{E}}\int_{0}^{T}\!\!\int_{D}\gamma_{\lambda}(\nabla u_{\lambda})\cdot\nabla u_{\lambda}<N.$$

$$\displaystyle\mathop{{}\mathbb{E}}\int_{0}^{T}\!\!\int_{D}k\bigl{(}(I+\lambda\gamma)^{-1}\nabla u_{\lambda}\bigr{)}\leq\mathop{{}\mathbb{E}}\int_{0}^{T}\!\!\int_{D}\gamma_{\lambda}(\nabla u_{\lambda})\cdot\nabla u_{\lambda}<N,$$

$$u_{\lambda }$$

$${\gamma }_{\lambda }(\nabla u_{\lambda })$$

$$\lambda u_{\lambda }$$

$$\lambda {\int }_0^t\Delta u_{\lambda }(s)ds⟶0\text{ in }{L}_{\omega }^2{H}^{-1},$$

$$E{\int }_0^t{\int }_D{\gamma }_{\lambda }(\nabla u_{\lambda }(s))\cdot \nabla \phi dxds⟶E{\int }_0^t{\int }_D\eta (s)\cdot \nabla \phi dxds,$$

$$E⟨u_{\lambda }(t),\phi ⟩⟶E⟨u(t),\phi ⟩.$$

$$u(t)-{\int }_0^t\mathrm{div}\eta (s)ds=u_0+{\int }_0^{t}G(s)dW(s)\text{in }{L}_{\omega }^1{V}_0^{\prime },$$

$$\mathop{{}\mathbb{E}}\int_{0}^{T}\!\!\int_{D}\bigl{(}k(\nabla u)+k^{*}(\eta)\bigr{)}<\infty.$$

$$y(t)+\alpha {\int }_0^{t}y(s)ds-{\int }_0^t\mathrm{div}\zeta (s)ds=y_0+{\int }_0^{t}C(s)dW(s)$$

Full text

1 Introduction and main result

Let us consider the stochastic partial differential equation

[TABLE]

posed on $L^{2}(D)$, with $D$ a bounded domain of $\mathbb{R}^{n}$ with smooth boundary. The following assumptions will be in force: (a) $\gamma$ is the subdifferential of a lower semicontinuous convex function $k:\mathbb{R}^{n}\to\mathbb{R}_{+}$ with $k(0)=0$ and such that

[TABLE]

(in particular, $\gamma$ is a maximal monotone graph in $\mathbb{R}^{n}\times\mathbb{R}^{n}$ whose domain coincides with $\mathbb{R}^{n}$); (b) $W$ is a cylindrical Wiener process on a separable Hilbert space $H$, supported by a filtered probability space $(\Omega,\mathscr{F},(\mathscr{F}_{t})_{t\in[0,T]},\mathbb{P})$ satisfying the “usual conditions”; (c) $B$ is a map from $\Omega\times[0,T]\times L^{2}(D)$ to $\mathscr{L}^{2}(H,L^{2}(D))$, the space of Hilbert-Schmidt operators from $H$ to $L^{2}(D)$, that is Lipschitz-continuous and has linear growth with respect to its third argument, uniformly with respect to the other two, and is such that $B(\cdot,\cdot,a)$ is measurable and adapted for all $a\in L^{2}(D)$.

Under the additional assumption that $\gamma$ is a (single-valued) continuous function, we proved in [7] that (1) admits a strong solution $u$, which is unique within a set of processes satisfying mild integrability conditions. The solution of [7] is constructed pathwise, i.e. for each $\omega\in\Omega$, so that, as is natural to expect, measurability problems arise with respect to the usual $\sigma$-algebras on $\Omega\times[0,T]$ used in the theory of stochastic processes. Precisely because of such an issue we needed to assume $\gamma$ to be single-valued.

The purpose of this note is to provide an alternative approach to establish the well-posedness of (1) that, avoiding pathwise constructions, is simpler than that of [7] and does not need any extra assumption on $\gamma$. The price to pay is that the solution we obtain here is less regular than that of [7]. We also refer to [9] for a related result obtained by analogous methods.

Let us define the concept of solution to (1) we shall be working with.

Definition 1.1**.**

Let $u_{0}$ be an $L^{2}(D)$-valued $\mathscr{F}_{0}$-measurable random variable. A strong solution to equation (1) is a couple $(u,\eta)$ satisfying the following properties:

• (i)

$u$* is a measurable and adapted $L^{2}(D)$-valued process such that*

[TABLE]

• (ii)

$\eta$* is a measurable and adapted $L^{1}(D)^{n}$-valued process such that*

[TABLE]

• (iii)

one has, as an equality in $L^{2}(D)$,

[TABLE]

Note that (2) has to be intended in the sense of distributions. In particular, since $\eta\in L^{1}(D)^{n}$, the integrand in the second term of (2) does not, in general, take values in $L^{2}(D)$. However, the conditions on $B$ imply that the stochastic integral in (2) is an $L^{2}(D)$-valued local martingale, hence the term involving the divergence of $\eta$ turns out to be $L^{2}(D)$-valued by comparison.

We can now state our main result. Here and in the following $k^{*}:\mathbb{R}^{n}\to\mathbb{R}_{+}$ is the convex conjugate of $k$, defined as k^{*}(y):=\sup_{x\in\mathbb{R}^{n}}\bigl{(}x\cdot y-k(x)\bigr{)}.

Theorem 1.2**.**

Let $u_{0}\in L^{2}(\Omega,\mathscr{F}_{0};L^{2}(D))$. Then equation (1) admits a unique strong solution $(u,\eta$) such that

[TABLE]

Moreover, the solution map $u_{0}\mapsto u$ is Lipschitz-continuous from $L^{2}(\Omega;L^{2}(D))$ to $L^{\infty}(0,T;L^{2}(\Omega;L^{2}(D)))$, and $u$ is weakly continuous as a function on $[0,T]$ with values in $L^{2}(\Omega;L^{2}(D))$.

Under the extra assumption of $\gamma$ being single-valued, the solution obtained in [7] is more regular in the sense that $\mathop{{}\mathbb{E}}\sup_{t\leq T}\lVert{u(t)}\rVert^{2}_{L^{2}(D)}$ is finite, the solution map is Lipschitz-continuous from $L^{2}(\Omega;L^{2}(D))$ to $L^{2}(\Omega;L^{\infty}(0,T;L^{2}(D)))$, and $u(\omega,\cdot)$ is weakly continuous as a function on $[0,T]$ with values in $L^{2}(D)$ for $\mathbb{P}$-a.a. $\omega\in\Omega$.

Acknowledgements. The authors are partially supported by The Royal Society through its International Exchange Scheme. Parts of this chapter were written while the first-named author was visiting the Interdisziplinäres Zentrum für Komplexe Systeme at the University of Bonn, hosted by Prof. S. Albeverio.

2 Well-posedness of an auxiliary equation

The goal of this section is to prove well-posedness of a version of (1) with additive noise. Namely, we consider the initial value problem

[TABLE]

where $G\in L^{2}(\Omega\times[0,T];\mathscr{L}^{2}(H,L^{2}(D)))$ is a measurable and adapted process.

Proposition 2.1**.**

Equation (3) admits a unique strong solution $(u,\eta)$ satisfying the same integrability and weak continuity conditions of Theorem 1.2.

We introduce the regularized equation

[TABLE]

where $\gamma_{\lambda}:\mathbb{R}^{n}\to\mathbb{R}^{n}$, $\gamma_{\lambda}:=\frac{1}{\lambda}(I-(I+\lambda\gamma)^{-1})$, for any $\lambda>0$, is the Yosida approximation of $\gamma$, and $\Delta:H^{1}_{0}(D)\to H^{-1}(D)$ is the (variational) Dirichlet Laplacian. Since $\gamma_{\lambda}$ is monotone and Lipschitz-continuous, the classical variational approach (see [4, 8] as well as [5]) yields the existence of a unique predictable process $u_{\lambda}$ with values in $H^{1}_{0}(D)$ such that

[TABLE]

and

[TABLE]

$\mathbb{P}$-a.s. in $H^{-1}(D)$ for all $t\in[0,T]$.

We are now going to prove a priori estimates and weak compactness in suitable topologies for $u_{\lambda}$ and related processes. These will allow us to pass to the limit as $\lambda\to 0$ in (4).

For notational parsimony, we shall often write, for any $p\geq 0$, $L^{p}_{\omega}$, $L^{p}_{t}$, and $L^{p}_{x}$ in place of $L^{p}(\Omega)$, $L^{p}(0,T)$, and $L^{p}(D)$, respectively, and $C_{t}$ to denote $C([0,T])$. Other similar abbreviations are self-explanatory. The $L^{2}(D)$-norm will be denoted simply by $\lVert{\cdot}\rVert$. If a function $f:D\to\mathbb{R}^{n}$ is such that each component $f^{j}$, $j=1,\ldots,n$, belongs to $L^{p}(D)$, we shall just write $f\in L^{p}(D)$ rather than $f\in L^{p}(D)^{n}$. The notation $a\lesssim b$ means that $a\leq Nb$ for a positive constant $N$.

Lemma 2.2**.**

There exists a constant $N$ such that

[TABLE]

Proof.

Itô’s formula for the square of the norm in $L^{2}_{x}$ yields

[TABLE]

hence, taking the supremum in time and expectation,

[TABLE]

where, by Davis’ inequality (see, e.g., [6]), the ideal property of Hilbert-Schmidt operators (see, e.g., [1, p. V.52]), and the elementary inequality $ab\leq\varepsilon a^{2}+b^{2}/\varepsilon$ $\forall a,b\geq 0$, $\varepsilon>0$,

[TABLE]

for any $\varepsilon>0$. To conclude it suffices to choose $\varepsilon$ small enough. ∎

Lemma 2.3**.**

The families $(\nabla u_{\lambda})$ and $(\gamma_{\lambda}(\nabla u_{\lambda}))$ are relatively weakly compact in $L^{1}(\Omega\times(0,T)\times D)$.

Proof.

Recall that, for any $y$, $r\in\mathbb{R}^{n}$, ones has $k(y)+k^{*}(r)=r\cdot y$ if and only if $r\in\partial k(y)=\gamma(y)$. Therefore, since

[TABLE]

we deduce by the definition of $\gamma_{\lambda}$ that

[TABLE]

(See, e.g., [3] for all necessary facts from convex analysis used in this note.) Hence, taking Lemma 2.2 into account, there exists a constant $N>0$, independent of $\lambda$, such that

[TABLE]

The assumptions on $k$ imply that its convex conjugate $k^{*}$ is also convex, lower semicontinuous and such that $\lim_{\lvert y\rvert\to\infty}k^{*}(y)/\lvert y\rvert=+\infty$. Therefore a simple modification of the criterion by de la Vallée Poussin implies that $(\gamma_{\lambda}(\nabla u_{\lambda}))$ is uniformly integrable on $\Omega\times(0,T)\times D$, hence that it is relatively weakly compact in $L^{1}_{t,\omega,x}$ by the Dunford-Pettis theorem. A completely analogous argument shows that

[TABLE]

hence that $(I+\lambda\gamma)^{-1}\nabla u_{\lambda}$ is relatively weakly compact in $L^{1}_{t,\omega,x}$. Moreover, since $\nabla u_{\lambda}=(I+\lambda\gamma)^{-1}\nabla u_{\lambda}+\lambda\gamma_{\lambda}(\nabla u_{\lambda})$, it also follows that $(\nabla u_{\lambda})$ is relatively weakly compact in $L^{1}_{t,\omega,x}$. ∎

Thanks to Lemmata 2.2 and 2.3, there exists a subsequence of $\lambda$, denoted by the same symbol, and processes $u\in L^{\infty}_{t}L^{2}_{\omega,x}\cap L^{1}_{t,\omega}W^{1,1}_{0}$ and $\eta\in L^{1}_{t,\omega,x}$ such that

[TABLE]

as $\lambda\to 0$. Let $t\in[0,T]$ be arbitrary but fixed. The fourth convergence above implies

[TABLE]

while the third yields, for any $\varphi\in L^{\infty}_{\omega}W^{1,\infty}$,

[TABLE]

hence $\displaystyle\mathop{{}\mathbb{E}}\int_{0}^{t}\langle\mathop{\mathrm{div}}\gamma_{\lambda}(\nabla u_{\lambda}(s)),\varphi\rangle\,ds\longrightarrow\mathop{{}\mathbb{E}}\int_{0}^{t}\langle\mathop{\mathrm{div}}\eta(s),\varphi\rangle\,ds$. Therefore, recalling (4), by difference we deduce that

[TABLE]

Consequently, since $u_{\lambda}(t)$ is bounded in $L^{2}_{\omega}L^{2}_{x}$, we also have that $u_{\lambda}(t)\rightarrow u(t)$ weakly in $L^{2}_{\omega}L^{2}_{x}$. Taking the limit as $\lambda\to 0$ in (4) thus yields

[TABLE]

where $V_{0}^{\prime}$ is the (topological) dual of a separable Hilbert space $V_{0}$ embedded continuously and densely in $H^{1}_{0}$, and continuously in $W^{1,\infty}$. The identity immediately implies that $u\in C_{t}L^{1}_{\omega}V_{0}^{\prime}$. Since $u\in L^{\infty}_{t}L^{2}_{\omega}L^{2}_{x}$, it follows by a result of Strauss (see [10, Theorem 2.1]) that $u$ is a weakly continuous function on $[0,T]$ with values in $L^{2}_{\omega}L^{2}_{x}$.

By Mazur’s lemma there exist sequences of convex combinations of $\gamma_{\lambda}(\nabla u_{\lambda})$ that converge $\eta$ in (the norm topology of) $L^{1}_{t,\omega,x}$, thus also, passing to a subsequence, $\mathbb{P}\otimes dt$-almost everywhere in $L^{1}_{x}$. Similarly, since $u_{\lambda}\to u$ weakly* in $L^{\infty}_{t}L^{2}_{\omega,x}$ implies that $u_{\lambda}\to u$ weakly in $L^{2}_{t,\omega,x}$, there exist sequences of convex combinations of $u_{\lambda}$ that converge to $u$ $\mathbb{P}\otimes dt$-almost everywhere in $L^{2}_{x}$. Since convex combinations of $(u_{\lambda})$ and of $(\gamma_{\lambda}(\nabla u_{\lambda}))$ are (at least) predictable and adapted, respectively, it follows that $u$ is predictable and $\eta$ is measurable and adapted. Moreover, thanks to the weak lower semicontinuity of convex integrals, one has

[TABLE]

In order to show that $\eta\in\gamma(\nabla u)$ for a.a. $(\omega,t,x)$, we need the following “energy identity”.

Lemma 2.4**.**

Assume that

[TABLE]

in $L^{2}_{x}$ $\mathbb{P}$-a.s. for all $t\in[0,T]$, where $\alpha\in\mathbb{R}$, $y_{0}\in L^{2}_{\omega,x}$ is $\mathscr{F}_{0}$-measurable, and

[TABLE]

are measurable and adapted processes such that $k(c\nabla y)+k^{*}(c\zeta)\in L^{1}_{t,\omega,x}$ for a constant $c>0$. Then

[TABLE]

Proof.

Let $m\in\mathbb{N}$ be such that such that $(I-\delta\Delta)^{-m}$ maps $L^{1}_{x}$ into $H^{1}_{0}\cap W^{1,\infty}$, and use the notation $h^{\delta}:=(I-\delta\Delta)^{-m}h$ for any $h$ taking values in $L^{1}_{x}$. One has

[TABLE]

$\mathbb{P}$-a.s. for all $t\in[0,T]$, as an equality in $L^{2}_{x}$, for which Itô’s formula yields

[TABLE]

It is evident from (6) that $y^{\delta}$ is a continuous $L^{2}_{x}$-valued process, hence the stochastic integral $(y^{\delta}C^{\delta})\cdot W$ on the right-hand side of the above identity is a continuous local martingale. Let $(T_{n})$ be a localizing sequence, and multiply the previous identity by $1_{[\!\![0,T_{n}]\!\!]}$, to obtain, thanks to $\mathop{{}\mathbb{E}}(y^{\delta}C^{\delta})\cdot W(\cdot\wedge T_{n})=0$,

[TABLE]

Letting $n$ tend to $\infty$, the dominated convergence theorem yields

[TABLE]

for all $t\in[0,T]$. We are now going to pass to the limit as $\delta\to 0$: the first and second terms on the left-hand side and the first on the right-hand side clearly converge to $\mathop{{}\mathbb{E}}\lVert{y(t)}\rVert^{2}$, $2\alpha\mathop{{}\mathbb{E}}\int_{0}^{t}\lVert{y(s)}\rVert^{2}\,ds$ and $\mathop{{}\mathbb{E}}\lVert{y_{0}}\rVert^{2}$, respectively. Properties of Hilbert-Schmidt operators and the dominated convergence theorem also yield

[TABLE]

for all $t\in[0,T]$. To conclude it then suffices to show that $\nabla y^{\delta}\cdot\zeta^{\delta}\to\nabla y\cdot\zeta$ in $L^{1}_{t,\omega,x}$. Since $\nabla y^{\delta}\to\nabla y$ and $\zeta^{\delta}\to\zeta$ in measure in $\Omega\times(0,t)\times D$, Vitali’s theorem implies strong convergence in $L^{1}_{t,\omega,x}$ if the sequence $(\nabla y^{\delta}\cdot\zeta^{\delta})$ is uniformly integrable in $\Omega\times(0,t)\times D$. In turn, the latter is certainly true if $(\lvert\nabla y^{\delta}\cdot\zeta^{\delta}\rvert)$ is dominated by a sequence that converges strongly in $L^{1}_{t,\omega,x}$. Indeed, using the assumptions on the behavior of $k$ at infinity as well as the generalized Jensen inequality for sub-Markovian operators (see [2]), one has

[TABLE]

where the sequence on the right-hand side converges in $L^{1}_{t,\omega,x}$ as $\delta\to 0$ because, by assumption, $k(c\nabla y)+k^{*}(c\zeta)\in L^{1}_{t,\omega,x}$. ∎

Itô’s formula yields

[TABLE]

and, by Lemma 2.4,

[TABLE]

One then has

[TABLE]

Since $\nabla u_{\lambda}\to\nabla u$ and $\gamma_{\lambda}(\nabla u_{\lambda})\to\eta$ weakly in $L^{1}_{t,\omega,x}$, this implies that $\eta\in\gamma(\nabla u)$ a.e. in $\Omega\times(0,T)\times D$. We have thus proved the existence and weak continuity statements of Proposition 2.1.

In order to show that the solution is unique, we are going to prove that any solution depends continuously on $(u_{0},G)$. Let $(u_{i},\eta_{i})$, $i=1,2$, satisfy

[TABLE]

in the sense of Definition 1.1, as well as the integrability conditions (on $u$ and $\eta$) of Theorem 1.2. Setting $y:=u_{1}-u_{2}$, $y_{0}:=u_{01}-u_{02}$, $\zeta:=\eta_{1}-\eta_{2}$, and $F:=G_{1}-G_{2}$, one has

[TABLE]

$\mathbb{P}$-a.s. in $L^{2}(D)$ for all $t\in[0,T]$. For any process $h$, let us use the notation $h^{\alpha}(t):=e^{-\alpha t}h(t)$. For any $\alpha>0$, the integration-by-parts formula yields

[TABLE]

hence also, thanks to Lemma 2.4,

[TABLE]

where $\zeta^{\alpha}\cdot\nabla y^{\alpha}\geq 0$ by monotonicity. Therefore, taking the $L^{\infty}_{t}$ norm,

[TABLE]

that is, using the notation $L^{p}_{t}(\alpha):=L^{p}([0,T],e^{-\alpha t}dt)$ for any $p\geq 0$,

[TABLE]

Taking $\alpha=0$ and $G_{1}=G_{2}$ immediately yields the uniqueness of solutions (as well as Lipschitz-continuous dependence on the initial datum). The proof of Proposition 2.1 is thus complete.

3 Proof of Theorem 1.2

For any $v\in L^{2}_{t,\omega,x}$ measurable and adapted, and any $\mathscr{F}_{0}$-measurable random variable $u_{0}\in L^{2}_{\omega,x}$, the process $B(\cdot,v)$ is measurable, adapted, and belongs to $L^{2}_{t,\omega}\mathscr{L}^{2}(H,L^{2}_{x})$, hence the equation

[TABLE]

is well-posed in the sense of Proposition 2.1. Moreover, for any $v_{1}$, $v_{2}$ and $u_{01}$, $u_{02}$ satisfying the same hypotheses on $v$ and $u_{0}$, respectively, (7) yields

[TABLE]

It hence follows by the Lipschitz-continuity of $B$ that

[TABLE]

where the implicit constant does not depend on $\alpha$. In particular, denoting by $\Gamma$ the map $(u_{0},v)\mapsto u$, one has that $v\mapsto\Gamma(u_{0},v)$ is a strict contraction of $L^{2}_{t}(\alpha)L^{2}_{\omega,x}$ for $\alpha$ large enough. Therefore, by equivalence of norms, $v\mapsto\Gamma(u_{0},v)$ admits a unique fixed point in $L^{2}_{t,\omega,x}$, which solves (1) and satisfies all integrability conditions. Such solution is unique as any solution is a fixed point of $v\mapsto\Gamma(u_{0},v)$.

Let us show that the solution map $u_{0}\mapsto u$ is Lipschitz-continuous: (8) yields, choosing $\alpha$ large enough,

[TABLE]

with constants $N_{1}>0$ and $0<N_{2}<1$, hence, by equivalence of norms,

[TABLE]

This in turn implies, in view of (7) (with $\alpha=0$) and the Lipschitz-continuity of $B$,

[TABLE]

which completes the proof.

Remark. A priori estimates entirely analogous to those of Lemma 2.2, as well as weak compactness results exactly as in Lemma 2.3, can be proved for the regularized equation obtained by replacing $\gamma$ with $\gamma_{\lambda}+\lambda\nabla$ directly in (1). It is however not immediately clear how to pass to the limit as $\lambda\to 0$ in the stochastic integrals appearing in such regularized equations with multiplicative noise, i.e. to show that $B(u_{\lambda})\cdot W$ converges to $B(u)\cdot W$ in a suitable sense.