Stochastic Generalized Porous Media Equations Over $\sigma$-finite Measure Spaces with Non-continuous Diffusivity Function

Abstract

In this paper, we prove that stochastic porous media equations over $\sigma$-finite measure spaces $(E,\mathcal {B},\mu )$, driven by time-dependent multiplicative noise, with the Laplacian replaced by a self-adjoint transient Dirichlet operator L and the diffusivity function given by a maximal monotone multi-valued function $\Psi$ of polynomial growth, have a unique solution. This generalizes previous results in that we work on general measurable state spaces, allow non-continuous monotone functions $\Psi$, for which, no further assumptions (as e.g. coercivity) are needed, but only that their multi-valued extensions are maximal monotone and of at most polynomial growth. Furthermore, an $L^p(\mu )$-Itô formula in expectation is proved, which is not only crucial for the proof of our main result, but also of independent interest. The result in particular applies to fast diffusion stochastic porous media equations (in particular self-organized criticality models) and cases where E is a manifold or a fractal, and to non-local operators L, as e.g. $L=-f(-\Delta )$, where f is a Bernstein function.

Appendix

1.1 A.1   Auxiliary Results

In this part we aim to prove Eq. A.4, which has been used in the proof of Claims 4.1 and 4.2.

Lemma A.1

Let $\nu ,{\varepsilon ,\lambda } \in (0,1]$. For all $x\in F^*_{1,2}$, we have

$$\begin{aligned}{} & {} \big \langle (\nu -L)((\Psi _\lambda +\lambda I)(J_\varepsilon (x))), x\big \rangle _{F^*_{1,2,\nu }}\nonumber \\= & {} \big \langle (\Psi _\lambda +\lambda I)(J_\varepsilon (x)),J_\varepsilon (x)\big \rangle _2+\varepsilon \Vert (\nu -L)((\Psi _\lambda +\lambda I)(J_\varepsilon (x)))\Vert ^2_{F^*_{1,2,\nu }}. \end{aligned}$$

(A.1)

For all $x\in L^2(\mu )$,

$$\begin{aligned}{} & {} \big \langle (\nu -L)(\Psi _\lambda +\lambda I)(J_\varepsilon (x)),x\big \rangle _2\nonumber \\= & {} \big \langle (\nu -L)(\Psi _\lambda +\lambda I)(J_\varepsilon (x)),J_\varepsilon (x)\big \rangle _2+\varepsilon \big |(\nu -L)(\Psi _\lambda +\lambda I)(J_\varepsilon (x))\big |^2_2. \end{aligned}$$

(A.2)

Proof

Recall from Eq. 4.4 that

$$\begin{aligned} J_\varepsilon (x)+\varepsilon (\nu -L)\big ((\Psi _\lambda +\lambda I)(J_\varepsilon (x))\big )=x,~~\forall x\in F_{1,2}. \end{aligned}$$

For $x\in F^*_{1,2}$, to prove Eq. A.1, we rewrite

$$\begin{aligned}{} & {} \big \langle (\nu -L)((\Psi _\lambda +\lambda I)(J_\varepsilon (x))), x\big \rangle _{F^*_{1,2,\nu }}\nonumber \\= & {} \big \langle (\nu -L)((\Psi _\lambda +\lambda I)(J_\varepsilon (x))), J_\varepsilon (x)\big \rangle _{F^*_{1,2,\nu }}\nonumber \\{} & {} +\big \langle (\nu -L)((\Psi _\lambda +\lambda I)(J_\varepsilon (x))), \varepsilon (\nu -L)\big ((\Psi _\lambda +\lambda I)(J_\varepsilon (x))\big )\big \rangle _{F^*_{1,2,\nu }}\nonumber \\= & {} \big \langle (\Psi _\lambda +\lambda I)(J_\varepsilon (x)),J_\varepsilon (x\big \rangle _2+\varepsilon \Vert (\nu -L)((\Psi _\lambda +\lambda I)(J_\varepsilon (x)))\Vert ^2_{F^*_{1,2,\nu }}. \end{aligned}$$

The proof of Eq. A.2 is analogous due to the fact that $J_\varepsilon$ is $\frac{1}{\sqrt{\nu \varepsilon \lambda }}$-Lipschitz in $L^2(\mu )$, so $A^{\nu ,\varepsilon }_\lambda \in L^2(\mu )$ if $x\in L^2(\mu )$.

Lemma A.2

Let $x\in L^2(\mu )$. Then for $\nu ,\varepsilon ,\lambda \in (0,1]$, $t\in [0,T]$, we have

$$\begin{aligned} \mathbb {E}|X^{\nu ,\varepsilon }_\lambda (t)|^2_2\!+2\mathbb {E}\!\!\int _0^t\!\!\big \langle (\nu -L)(\Psi _\lambda +\lambda I)(J_\varepsilon (X^{\nu ,\varepsilon }_\lambda (s))),J_\varepsilon (X^{\nu ,\varepsilon }_\lambda (s))\big \rangle _2ds\le e^{C_3T}|x|^2_2. \end{aligned}$$

(A.3)

Proof

Applying Itô’s formula to $|X^{\nu ,\varepsilon }_\lambda |^2_2$, we obtain

$$\begin{aligned}{} & {} d|X^{\nu ,\varepsilon }_\lambda (t)|^2_2+2\big \langle (\nu -L)((\Psi _\lambda +\lambda I)(J_\varepsilon (X^{\nu ,\varepsilon }_\lambda (t)))),(X^{\nu ,\varepsilon }_\lambda (t)\big \rangle _2dt\nonumber \\= & {} \Vert B(t,X^{\nu ,\varepsilon }_\lambda (t))\Vert _{L_2(L^2(\mu ),L^2(\mu ))}^2dt+2\big \langle X^{\nu ,\varepsilon }_\lambda ,B(t,X^{\nu ,\varepsilon }_\lambda (t))dW(t)\big \rangle _2 \end{aligned}$$

which by Eq. A.2 yields,

$$\begin{aligned}{} & {} d|X^{\nu ,\varepsilon }_\lambda (t)|^2_2+2\big \langle (\nu -L)((\Psi _\lambda +\lambda I)(J_\varepsilon (X^{\nu ,\varepsilon }_\lambda (t)))), J_\varepsilon (X^{\nu ,\varepsilon }_\lambda (t))\big \rangle _2dt\nonumber \\{} & {} +2\varepsilon \big |(\nu -L)((\Psi _\lambda +\lambda I)(J_\varepsilon (X^{\nu ,\varepsilon }_\lambda (t))))\big |^2_2dt\nonumber \\= & {} \Vert B(t,X^{\nu ,\varepsilon }_\lambda (t))\Vert _{L_2(L^2(\mu ),L^2(\mu ))}^2dt+2\big \langle X^{\nu ,\varepsilon }_\lambda ,B(t,X^{\nu ,\varepsilon }_\lambda (t))dW(t)\big \rangle _2. \end{aligned}$$

Taking expectation of both sides, by (H3)(i) we get

$$\begin{aligned}{} & {} \mathbb {E}|X^{\nu ,\varepsilon }_\lambda (t)|^2_2+2\mathbb {E}\int _0^t\big \langle (\nu -L)((\Psi _\lambda +\lambda I)(J_\varepsilon (X^{\nu ,\varepsilon }_\lambda (s)))) ,J_\varepsilon (X^{\nu ,\varepsilon }_\lambda (s))\big \rangle _2ds\nonumber \\{} & {} +2\varepsilon \mathbb {E}\int _0^t\big |(\nu -L)((\Psi _\lambda +\lambda I)(J_\varepsilon (X^{\nu ,\varepsilon }_\lambda (s))))\big |_2^2ds\nonumber \\\le & {} |x|^2_2+C_3\mathbb {E}\int _0^t|X^{\nu ,\varepsilon }_\lambda (s)|^2_2ds. \end{aligned}$$

Then by Eq. 4.16 and Gronwall’s lemma we get Eq. A.3 as claimed.

Proposition A.1

Let $x\in L^2(\mu )$. Then for $\nu ,\varepsilon ,\lambda \in (0,1]$, $t\in [0,T]$, we have

$$\begin{aligned} \mathbb {E}\int _0^t\big \Vert (\nu -L)\big ((\Psi _\lambda +\lambda I)(J_\varepsilon (X^{\nu ,\varepsilon }_\lambda (s)))\big )\big \Vert ^2_{F^*_{1,2,\nu }}ds\le \frac{1}{2}(\frac{1}{\lambda }+\lambda +C_5)e^{C_3T}|x|^2_2. \end{aligned}$$

(A.4)

Proof

Let $x\in L^2(\mu )$. Then

$$\begin{aligned}{} & {} \big \Vert (\nu -L)\big ((\Psi _\lambda +\lambda I)(J_\varepsilon (x))\big )\big \Vert ^2_{F^*_{1,2,\nu }}\nonumber \\= & {} \Vert (\Psi _\lambda +\lambda I)(J_\varepsilon (x))\Vert ^2_{F_{1,2,\nu }}\nonumber \\= & {} \int \frac{1}{2}\Gamma \big ((\Psi _\lambda +\lambda I)(J_\varepsilon (x)),(\Psi _\lambda +\lambda I)(J_\varepsilon (x))\big )d\mu \nonumber \\{} & {} + \nu \big \langle (\Psi _\lambda +\lambda I)(J_\varepsilon (x)),(\Psi _\lambda +\lambda I)(J_\varepsilon (x))\big \rangle _2\nonumber \\\le & {} C_5\int \frac{1}{2}\Gamma \big (J_\varepsilon (x),(\Psi _\lambda +\lambda I)(J_\varepsilon (x))\big )d\mu +\nu (\frac{1}{\lambda }+\lambda )\langle (\Psi _\lambda +\lambda I)(J_\varepsilon (x)),J_\varepsilon (x)\rangle _2\nonumber \\\le & {} (\frac{1}{\lambda }+\lambda +C_5)\langle J_\varepsilon (x),(\Psi _\lambda +\lambda I)(J_\varepsilon (x))\rangle _{F_{1,2,\nu }}\nonumber \\= & {} (\frac{1}{\lambda }+\lambda +C_5)\langle (\nu -L)(\Psi _\lambda +\lambda I)(J_\varepsilon (x)),J_\varepsilon (x)\rangle _2, \end{aligned}$$

where in the first inequality we used (H4), the fact that $r(\Psi _\lambda (r)+\lambda r)\ge 0$ for all $r\in \mathbb {R}$ and $\Psi _\lambda$ is $\frac{1}{\lambda }$-Lipschitz ([3, Page:41, Proposition 2.3 (ii)]), the last equality comes from the fact that $(\Psi _\lambda +\lambda I)(J_\varepsilon (x))\in D(L)$. Now from Eq. A.3, we get the assertion.

1.2 A.2   The $L^p$-Itô Formula in Expectation

The purpose in this section is to prove Theorem 7.1 below, which has been used in Lemmas 4.2 and 4.3.

Let $\ell _2$ be the space of all square-summable sequences in $\mathbb {R}$ and $p\in [2,\infty )$. In addition, to the real-valued $L^p$-space, $L^p(\mu ):=L^p(E,\mu )$ we consider the $\ell _2$-valued $L^p$-space $L^p(\mu ;\ell _2):=L^p(E,\mu ;\ell _2)$. We set

$$|g|_p^p:=|g|^p_{L^p(\mu ;\ell _2)}=\int _E\Vert g(x)\Vert ^p_{\ell _2}\mu (dx)=\int _E\Big (\sum _{k=1}^\infty |g_k(x)|^2\Big )^{\frac{p}{2}}\mu (dx).$$

Let $\mathscr {P}$ denote the predictable $\sigma$-algebra on $[0,T]\times \Omega$ corresponding to $(\Omega ,\mathscr {F},(\mathscr {F}_t)_{t\ge 0},\mathbb {P})$. For $p\in [2,\infty )$ we set

$$\begin{aligned} \mathbb {L}^p(T):=L^p([0,T]\times \Omega ,\mathscr {P};L^p(\mu )) \end{aligned}$$

and

$$\begin{aligned} \mathbb {L}^p(T;\ell _2):=L^p([0,T]\times \Omega ,\mathscr {P};L^p(\mu ;\ell _2)), \end{aligned}$$

equipped with their standard $L^p$-norms. Since $(E,\mathcal {B})$ is a standard measurable space, by definition there exists a complete metric d on E, such that (E, d) is separable, i.e., a Polish space, whose Borel $\sigma$-algebra coincides with $\mathcal {B}$. Below we fix this metric d and denote the corresponding set of all bounded continuous functions by $C_b(E)$.

Let $\mathcal {E}$ be all $g=(g_k)_{k\in \mathbb {N}}\in L^\infty ([0,T]\times \Omega ;L^\infty (\mu ;\ell _2)\cap L^1(\mu ;\ell _2))$ such that there exists $j\in \mathbb {N}$ and bounded stopping times $\tau _0\le \tau _1\le \cdots \le \tau _j\le T$ such that

$$\begin{aligned} g_k=\left\{ \begin{array}{ll} \sum _{i=1}^jg_k^i1_{(\tau _{i-1},\tau _i]}, &{} \textrm{if}~k\le j; \\ 0, &{} \textrm{if}~k>j, \end{array} \right. \end{aligned}$$

where $g_k^i\in C_b(E)\cap L^1(\mu )$, $1\le i\le j$.

Claim A.1

$\mathcal {E}$ is dense in $\mathbb {L}^p(T;\ell _2)$ for all $p\in [2,\infty )$.

Proof

Let $f=(f_k)_{k\in \mathbb {N}}\in L^q(T;\ell _2)$, with $q:=\frac{p}{p-1}$, be such that

$$\begin{aligned} _{\mathbb {L}^q(T;\ell _2)}\langle f,g\rangle _{\mathbb {L}^p(T;\ell _2)}=\mathbb {E}\int _{0}^{T}\int _E\sum _{k=1}^{\infty }f_kg_kd\mu ds=0 ~~\forall g\in \mathcal {E}. \end{aligned}$$

Now let $\sigma \le \tau$ be two stopping times and $k\in \mathbb {N}$. Define $g\in \mathbb {L}^p(T;\ell _2)$ by $g=(g_k\delta _{ik})_{i\in \mathbb {N}}$, where

$$g_k:=g_k^kI_{(\sigma ,\tau ]}$$

and $g_k^k\in C_b(E)\cap L^1(\mu )$. Then $g\in \mathcal {E}$, hence

$$\begin{aligned} 0= & {} _{\mathbb {L}^q(T;\ell _2)}\langle f,g\rangle _{\mathbb {L}^p(T;\ell _2)}\nonumber \\= & {} \mathbb {E}\int _0^T\int _Ef_kg_k^kd\mu \ I_{(\sigma ,\tau ]}(t)dt, \end{aligned}$$

which implies that

$$\begin{aligned} \int _Ef_kg_k^kd\mu =0~~dt\otimes \mathbb {P}-a.s., \end{aligned}$$

since all sets of the type $(\sigma ,\tau ]$ generate the $\sigma$-algebra $\mathscr {P}$ and since $f_k$ is $\mathscr {P}$-measurable. Therefore, since $C_b(E)\cap L^1(\mu )$ is dense in $L^p(\mu )$,

$$\begin{aligned} f_k=0~~\text {in}~~L^q(\mu )~~dt\otimes \mathbb {P}-a.s.,~~\text {for~all}~~k\in \mathbb {N}. \end{aligned}$$

Now the assertion follows by the Hahn-Banach theorem ([38, page: 61, Corollary 4.23]).

Remark A.1

Let $\mathcal {S}$ be the set of all functions $f\in L^\infty ([0,T]\otimes \Omega ;L^\infty (\mu )\cap L^1(\mu ))$ such that there exist $l\in \mathbb {N}$ and bounded stopping times $\tau '_0\le \tau '_1\le \cdots \le \tau '_l\le T$ such that $f=\sum _{i=1}^lf^i1_{(\tau '_{i-1},\tau '_i]}$, where $f^i\in C_b(E)\cap L^1(\mu )$, $1\le i\le l$. Similarly to Claim 7.1, one can prove that $\mathcal {S}$ is dense in $\mathbb {L}^p(T)$ for all $p\in [2,\infty )$.

Define $\textbf{M}:\mathcal {E}\longmapsto \bigcap _{p\ge 1}L^p(\Omega ;C([0,T];L^p(\mu )))$ as follows:

$$\begin{aligned} \textbf{M}(g)(t)=\int _0^tgdW(s){} & {} :=\sum _{k=1}^{\infty }\int _{0}^{t}g_kdW_k(s)\nonumber \\{} & {} \!=\!\sum _{i,k\!=\!1}^{j}g_k^i\big (W_k(t\wedge \tau _i)\!-\!W_k(t\wedge \tau _{i-1})\big ),~t\!\in \![0,T],~g\!\in \!\mathcal {E}. \end{aligned}$$

(A.5)

Let us note that the right hand-side of Eq. A.5 is $\mathbb {P}$-a.s. for every $t\in [0,T]$ a continuous $\mu$-version of $\textbf{M}(g)(t)\in L^p(E,\mu )$, which for every $x\in E$ is a continuous real-valued martingale and is equal to

$$\begin{aligned} \sum _{k=1}^\infty \int _0^tg_k(s,x)dW_k(s),~x\in E, ~t\in [0,T]. \end{aligned}$$

(A.6)

Claim A.2

Let $p\in [2,\infty )$. Then $\textbf{M}$ extends to a linear continuous map $\overline{\textbf{M}}$ from $\mathbb {L}^p(T;\ell _2)$ to $L^p(\Omega ;C([0,T];L^p(\mu )))$, such that $\overline{\textbf{M}}(g)$ is a continuous martingale in $L^p(\mu )$ for all $g\in \mathbb {L}^p(T;\ell _2)$.

Proof

We have

$$\begin{aligned}{} & {} \mathbb {E}\Big [\sup _{t\in [0,T]}\int _E\big |\int _0^tgdW(s)\big |^pd\mu \Big ]\nonumber \\= & {} \mathbb {E}\Big [\sup _{t\in [0,T]}\int _E\Big |\sum _{k=1}^{\infty }\int _{0}^{t}g_{k}(s,x)dW_k(s)\Big |^pd\mu \Big ]\nonumber \\\le & {} \int _E\Big [\mathbb {E}\sup _{t\in [0,T]}\Big |\sum _{k=1}^{\infty }\int _{0}^{t}g_{k}(s,x)dW_k(s)\Big |^p\Big ]d\mu \nonumber \\\le & {} c_p\int _E\Big [\mathbb {E}\Big \langle \sum _{k=1}^{\infty }\int _0^\cdot ~g_{k}(s,x)dW_k(s)\Big \rangle ^{\frac{p}{2}}_T\Big ]d\mu \nonumber \\= & {} c_p\int _E\mathbb {E}\Big (\sum _{k=1}^{\infty }\int _0^Tg^2_{k}(s,x)ds\Big )^{\frac{p}{2}}d\mu \nonumber \\= & {} c_p\mathbb {E}\Bigg [\int _E\Big (\int _{0}^{T}|g(s,x)|^2_{\ell _2}ds\Big )^{\frac{p}{2}}d\mu \Bigg ]^{\frac{2}{p}\cdot \frac{p}{2}}\nonumber \\\le & {} c_p\mathbb {E}\Bigg [\int _{0}^{T}\Big (\int _E|g(s,x)|^p_{\ell _2}d\mu \Big )^{\frac{2}{p}}ds\Bigg ]^{\frac{p}{2}}\nonumber \\\le & {} c_pT^{\frac{p}{2}-1}\mathbb {E}\int _{0}^{T}\big |g(s,\cdot )\big |^p_{L^p(\mu ;\ell _2)}ds, \end{aligned}$$

where we have used the BDG inequality applied to the real-valued martingale in Eq. A.6 in the third step, the assumption that $p\ge 2$ and Minkowski’s inequality in the sixth step and Hölder’s inequality in the last step. Hence the first part of the assertion follows.

To prove the second let $g\in \mathbb {L}^p(T;\ell _2)$. It suffices to prove that for all $f\in L^q(\mu )$ with $q:=\frac{p}{p-1}$,

$$\begin{aligned} \int _Ef~\overline{\textbf{M}}(g)(t)d\mu ,~t\in [0,T], \end{aligned}$$

is a real-valued martingale (see e.g. [26, Remark 2.2.5]). But since for some $g_n\in \mathcal {E}$, $n\in \mathbb {N}$, we have $\forall ~t\in [0,T]$ that

$$\begin{aligned} \textbf{M}(g_n)(t)~~\overrightarrow{n\rightarrow \infty }~~\overline{\textbf{M}}(g)(t)~~\text {in}~~L^p(\Omega ;L^p(\mu )), \end{aligned}$$

it follows that

$$\begin{aligned} \int _Ef~\textbf{M}(g_n)(t)d\mu ~~\overrightarrow{n\rightarrow \infty }~~\int _Ef~\overline{\textbf{M}}(g)(t)d\mu ~~\text {in}~~L^1(\Omega ). \end{aligned}$$

So, we may assume that $g\in \mathcal {E}$. But in this case by Eq. A.5 it follows immediately that $\int _Ef~\textbf{M}(g)(t)d\mu$, $t\in [0,T]$, is a real-valued martingale.

Below we define for $g\in \mathbb {L}^p(T;\ell _2)$, $p\in [2,\infty )$,

$$\begin{aligned} \int _0^tg(s)dW(s):=\overline{\textbf{M}}(g)(t),~~t\in [0,T], \end{aligned}$$

where $\overline{\textbf{M}}$ is as in Claim 7.2.

Now we fix $p\in [2,\infty )$ and consider the following process

$$u:\Omega \times [0,T]\rightarrow L^p(\mu ),$$

defined by

$$\begin{aligned} u(t):=u(0)+\int _0^tf(s)ds+ \int _0^tg(s)dW(s), \end{aligned}$$

(A.7)

where $u(0)\in L^p(\Omega ,\mathcal {F}_0;L^p(\mu ))$, $f\in \mathbb {L}^p(T)$ and $g\in \mathbb {L}^p(T;\ell _2)$.

Theorem A.1

“Itô-formula in expectation" Let $p\in [2,\infty )$, $f\in \mathbb {L}^p(T)$, $g\in \mathbb {L}^p(T;\ell _2)$. Let u be as in Eq. A.7. Then for all $t\in [0,T]$,

$$\begin{aligned} \mathbb {E}|u(t,x)|_p^p= & {} \mathbb {E}|u(0)|^p+\mathbb {E}\int _0^t\int _Ep|u(s,x)|^{p-2}u(s,x)f(s,x)\mu (dx) ds\nonumber \\{} & {} +\frac{1}{2}p(p-1)\mathbb {E}\int _0^t\int _E|u(s,x)|^{p-2} |g(s,x)|^2_{\ell _2}\mu (dx) ds. \end{aligned}$$

(A.8)

Remark A.2

In the case $E=\mathbb {R}^d$, $\mu =$Lebesgue measure, N. Krylov proved Itô’s formula for the $L^p$-norm of a large class of $W^{1,p}$-valued stochastic processes in his fundamental paper [25]. In particular, Lemma 5.1 in that paper gives a pathwise Itô formula for processes u as in Eq. A.7, which immediately implies Eq. A.8. The proof, however, uses a smoothing technique by convoluting the process u in x with Dirac-sequence of smooth functions, which is not available in our more general case, where $(E,\mathcal {B})$ is just a standard measurable space with a $\sigma$-finite measure $\mu$, without further structural assumptions that we wanted to avoid to cover applications e.g. to underlying spaces E which are fractals. Fortunately, the above Itô formula in expectation is enough to prove all main results in this paper without any further assumptions. After the preparations above, its proof is quite simple.

We recall the following well-known result (see e.g. Theorem 21.7 in [10]):

Lemma A.3

Let $p\in [1,\infty )$, $v_n,v\in L^p(\mu )$ such that $v_n\rightarrow v$ in $\mu$-measure as $n\rightarrow \infty$ and

$$\lim _{n\rightarrow \infty }|v_n|_p=|v|_p.$$

Then

$$\lim _{n\rightarrow \infty }v_n=v~~\text {in}~~L^p(\mu ).$$

Proof of Theorem 7.1

   By Claim 7.1 and Remark 7.1, we can find $f_n\in \mathcal {S}$, $n\in \mathbb {N}$, and $g_n\in \mathbb {L}^p(T;\ell _2)$, $n\in \mathbb {N}$, such that as $n\rightarrow \infty$

$$\begin{aligned} f_n\rightarrow f~~\text {in}~~\mathbb {L}^p(T), \end{aligned}$$

(A.9)

and

$$\begin{aligned} g_n\rightarrow g ~~\text {in}~~\mathbb {L}^p(T;\ell _2). \end{aligned}$$

(A.10)

For $n\in \mathbb {N}$, define

$$\begin{aligned} u_n(t):=u(0)+\int _0^tf_n(s)ds+\int _0^tg_n(s)dW(s). \end{aligned}$$

By Eqs. A.7, A.9, A.10 and Claim 7.2, it follows that as $n\rightarrow \infty$,

$$\begin{aligned} \int _0^\cdot f_n(s)ds{} & {} \rightarrow \int _0^\cdot f(s)ds,\nonumber \\ \int _0^\cdot g_n(s)dW(s){} & {} \rightarrow \int _0^\cdot g(s)dW(s),\\ u_n{} & {} \rightarrow u,\nonumber \end{aligned}$$

(A.11)

in $L^p(\Omega ;C([0,T];L^p(\mu )))$.

Applying the Itô formula to the real-valued semi-martingale $|u_n(t,x)|_p^p$ for each $x\in E$, and integrating w.r.t. $x\in E$ and $\omega \in \Omega$, we obtain

$$\begin{aligned} \mathbb {E}\int _E|u_n(t,x)|^p\mu (dx)= & {} \mathbb {E}|u(0)|^p+\mathbb {E}\int _E\int _0^tp|u_n(s,x)|^{p-2}u_n(s,x)\cdot f_n(s,x)ds\mu (dx)\nonumber \\{} & {} +\frac{1}{2}p(p-1)\mathbb {E}\int _E\int _0^t|u_n(s,x)|^{p-2}\cdot |g_n(s,x)|^2_{\ell _2}ds\mu (dx). \end{aligned}$$

(A.12)

Note that by Lemma 7.3 and Eq. A.11

$$\begin{aligned} |u_n(s)|^{p-2}u_n(s){} & {} \rightarrow |u(s)|^{p-2}u(s)~~\text {in}~~L^{\frac{p}{p-1}}(\mu ),\\ |u_n(s)|^{p-2}{} & {} \rightarrow |u(s)|^{p-2}~~\text {in}~~L^{\frac{p}{p-2}}(\mu ), \end{aligned}$$

as $n\rightarrow \infty$. Hence by Eqs. A.9 and A.10 we may pass to the limit $n\rightarrow \infty$ in Eq. A.12 to get Eq. A.8. $\square$