Let ${(X_t)}_{t\in [0,T]}$ be a continuous semi-martingale on a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ with values in a separable Hilbert space $(\mathcal{H},{\langle \cdot ,\cdot \rangle }_{\mathcal{H}})$ with basis ${(e_i)}_{i\in \mathbb{N}}$. Then its quadratic variation process is defined as

${\langle \langle X,X\rangle \rangle }_t^{\mathcal{H}}:=\sum _{i,j\in \mathbb{N}}{\langle \langle {\langle X,e_i\rangle }_{\mathcal{H}},{\langle X,e_j\rangle }_{\mathcal{H}}\rangle \rangle }_t{\langle e_j,\cdot \rangle }_{\mathcal{H}}{e}_i$

and has values in $\mathcal{N}(\mathcal{H})$ (the set of nuclear operators on $\mathcal{H}$). Moreover, we define the trace of ${\langle \langle X,X\rangle \rangle }_t^{\mathcal{H}}$ by

$\mathrm{tr}{\langle \langle X,X\rangle \rangle }_t^{\mathcal{H}}:=\sum _{i\in \mathbb{N}}{\langle \langle {\langle X,e_i\rangle }_{\mathcal{H}},{\langle X,e_i\rangle }_{\mathcal{H}}\rangle \rangle }_t{\langle e_i,\cdot \rangle }_{\mathcal{H}}{e}_i.$

Let $(\mathcal{H},{\langle \cdot ,\cdot \rangle }_{\mathcal{H}})$ be a separable Hilbert space, $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space and ${(X_t)}_{t\in [0,T]}$ be a continuous martingale on $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ with values in $\mathcal{H}$. Then we have for all $p>0$

$c_p\mathbb{E}{[\underset{t\in (0,T)}{\sup }{\Vert X_t\Vert }_{\mathcal{H}}]}^p⩽\mathbb{E}{\left[{\Vert \mathrm{tr}{\langle \langle X,X\rangle \rangle }_T^{\mathcal{H}}\Vert }_{\mathcal{N}(\mathcal{H})}\right]}^{\frac{p}{2}}⩽C_p\mathbb{E}{[\underset{t\in (0,T)}{\sup }{\Vert X_t\Vert }_{\mathcal{H}}]}^p,$

where $c_p,C_p$ are positive constants.

Let $(\mathcal{V},{\Vert \cdot \Vert }_{\mathcal{V}})$ be a Banach space which is continuously embedded into a separable Hilbert space $(\mathcal{H},{\langle \cdot ,\cdot \rangle }_{\mathcal{H}})$. Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space. Assume that the processes ${(X_t)}_{t\in [0,T]}$ and ${(Y_t)}_{t\in [0,T]}$, taking values in $\mathcal{V}$ and ${\mathcal{V}}^{\prime }$, respectively, are progressively measurable and

$\mathbb{P}\{{\int }_0^T({\Vert X\Vert }_{\mathcal{V}}^2+{\Vert Y\Vert }_{{\mathcal{V}}^{\prime }}^2)\mathrm{d}t<\infty \}=1.$

Assume further that there is a continuous martingale ${(M_t)}_{t\in [0,T]}$, taking values in $\mathcal{H}$, such that, for $\mathbb{P}\times {\mathcal{L}}^1$-a.e. $(\omega ,t)$, the following equality holds:

${\langle X(t),\phi \rangle }_{\mathcal{H}}={\langle X(0),\phi \rangle }_{\mathcal{H}}+{{\int }_0^t}_{{\mathcal{V}}^{\prime }}{\langle Y(\sigma ),\phi \rangle }_{\mathcal{V}}\mathrm{d}\sigma +{\langle M_t,\phi \rangle }_{\mathcal{H}}\forall \phi \in \mathcal{V}.$

Then we have

$\begin{array}{cc}\hfill {\Vert X(t)\Vert }_{\mathcal{H}}^2& ={\Vert X(0)\Vert }_{\mathcal{H}}^2+{{\int }_0^t}_{{\mathcal{V}}^{\prime }}{\langle Y(\sigma ),X(s)\rangle }_{\mathcal{V}}\mathrm{d}\sigma \hfill \\ \hfill & +2{\int }_0^t{\langle X(\sigma ),\mathrm{d}{M}_{\sigma }\rangle }_{\mathcal{H}}+{\Vert \mathrm{tr}{\langle \langle M,M\rangle \rangle }_t^{\mathcal{H}}\Vert }_{\mathcal{N}(\mathcal{H})},\mathbb{P}\times {\mathcal{L}}^1\mathit{\text{-a.e.}}\hfill \end{array}$

Let $\mathrm{\Psi }\in L^p(\mathrm{\Omega },\mathcal{F},\mathbb{P};L^p(0,T;L_2(U,L^2(G))))$ ($p⩾2$) be progressively ${({\mathcal{F}}_t)}_{t⩾0}$-measurable and W a cylindrical ${({\mathcal{F}}_t)}_{t⩾0}$-Wiener process as in (9.1.1). Then the following holds for any $\alpha \in (0,1/2)$

$\mathbb{E}\left[{\Vert {\int }_0^{\cdot }\mathrm{\Psi }\mathrm{d}{\mathbf{W}}_{\sigma }\Vert }_{W^{\alpha ,p}(0,T;L^2(G))}^p\right]⩽c(\alpha ,p)\mathbb{E}\left[{\int }_0^T{\Vert \mathrm{\Psi }\Vert }_{L_2(U,L^2(G))}^p\mathrm{d}t\right].$

Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space endowed with the filtration ${({\mathcal{F}}_t)}_{t⩾0}$.

a) Let W be a M-dimensional Brownian motion with respect to ${({\mathcal{F}}_t)}_{t⩾0}$. Let $\mathbf{X}\in L^{\beta }(\mathrm{\Omega },\mathcal{F},\mathbb{P};L^{\beta }(0,T))$, $\beta >2$, be a progressively ${({\mathcal{F}}_t)}_{t⩾0}$-measurable process with values in ${\mathbb{R}}^{N\times M}$. Then the paths of the process ${\mathbf{Z}}_t:={\int }_0^t\mathbf{X}\mathrm{d}{\mathbf{W}}_{\sigma }$ are $\mathbb{P}$-a.s. Hölder continuous with exponent $\alpha \in (\frac{1}{\beta },\frac{1}{2})$ and we have

$\mathbb{E}\left[{\Vert \mathbf{Z}\Vert }_{C^{\alpha }([0,T])}^{\beta }\right]⩽c_{\alpha }\mathbb{E}[{\int }_0^T|\mathbf{X}{|}^{\beta }\mathrm{d}t].$

b) Let $\psi \in L^{\beta }(\mathrm{\Omega },\mathcal{F},\mathbb{P};L^{\beta }(0,T;L_2(U,L^2(G))))$, $\beta >2$, be progressively ${({\mathcal{F}}_t)}_{t⩾0}$-measurable and W a cylindrical ${({\mathcal{F}}_t)}_{t⩾0}$-Wiener process as in (9.1.1). Then the paths of the process ${\mathbf{Z}}_t:={\int }_0^t\psi \mathrm{d}{\mathbf{W}}_{\sigma }$ are $\mathbb{P}$-a.s. Hölder continuous with exponent $\alpha \in (\frac{1}{\beta },\frac{1}{2})$ and the following holds

$\mathbb{E}\left[{\Vert \mathbf{Z}\Vert }_{C^{\alpha }([0,T];L^2(G))}^{\beta }\right]⩽c_{\alpha }\mathbb{E}\left[{\int }_0^T{\Vert \psi \Vert }_{L_2(U,L^2(G))}^{\beta }\mathrm{d}t\right].$

a) We follow [94], proof of Lemma 4.6, and consider the Riemann–Liouville operator: let X be a Banach space, $p\in (1,\infty ]$, $\alpha \in (\frac{1}{p},1]$ and $f\in L^p(0,T;X)$. Then the Riemann–Liouville operator is given by

$(R_{\alpha }f)(t):=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-\sigma )}^{\alpha -1}f(\sigma )\mathrm{d}\sigma ,t\in [0,T].$

It is well known that $R_{\alpha }$ is a bounded linear operator from $f\in L^p(0,T;X)$ to $C^{\alpha -1/p}([0,T];X)$ (see [130], Thm. 3.6). According to the stochastic Fubini Theorem (see [55], Thm. 4.18) we have ${\mathbf{Z}}_t=R_{\alpha }({\tilde{\mathbf{Z}}}_t)$, where

${\tilde{\mathbf{Z}}}_t=\frac{1}{\mathrm{\Gamma }(1-\alpha )}{\int }_0^t{(t-\sigma )}^{-\alpha }\mathbf{X}(t)\mathrm{d}{\mathbf{W}}_{\sigma },t\in [0,T].$

For $\alpha \in (\frac{1}{\beta },\frac{1}{2})$ we obtain by the Burkholder–Davis–Gundi inequality and Young's inequality for convolution that

$\begin{array}{cc}\hfill \mathbb{E}\left[{\Vert \mathbf{Z}\Vert }_{C^{\alpha -1/\beta }([0,T])}^{\beta }\right]& ⩽c\mathbb{E}[{\int }_0^T|\tilde{\mathbf{Z}}(t){|}^{\beta }\mathrm{d}t]⩽c{\int }_0^T\mathbb{E}{[\underset{[0,t]}{\sup }|\tilde{\mathbf{Z}}(\sigma )|]}^{\beta }\mathrm{d}t\hfill \\ \hfill & ⩽c{\int }_0^T\mathbb{E}{[{\int }_0^t{(t-\sigma )}^{-2\alpha }|\mathbf{X}(\sigma ){|}^2\mathrm{d}\sigma ]}^{\frac{\beta }{2}}\mathrm{d}t\hfill \\ \hfill & ⩽c\mathbb{E}[{\int }_0^T|\mathbf{X}{|}^{\beta }\mathrm{d}t].\hfill \end{array}$

b) By exactly the same arguments we end up with

$\begin{array}{cc}\hfill & \mathbb{E}\left[{\Vert \mathbf{Z}\Vert }_{C^{\alpha -1/\beta }([0,T];L^2(G))}^{\beta }\right]\hfill \\ \hfill & ⩽c\mathbb{E}\left[{\int }_0^T{\Vert \tilde{\mathbf{Z}}(t)\Vert }_{L^2(G)}^{\beta }\mathrm{d}t\right]⩽c{\int }_0^T\mathbb{E}{[\underset{[0,t]}{\sup }{\Vert \tilde{\mathbf{Z}}(\sigma )\Vert }_{L^2(G)}^{\beta }]}^{\beta }\mathrm{d}t\hfill \\ \hfill & ⩽c{\int }_0^T\mathbb{E}{\left[\sum _i{\int }_0^t{(t-\sigma )}^{-2\alpha }{\Vert \psi ({\mathbf{e}}_i)\Vert }_{L^2(G)}^2\mathrm{d}\sigma \right]}^{\frac{\beta }{2}}\mathrm{d}t\hfill \\ \hfill & =c{\int }_0^T\mathbb{E}{\left[{\int }_0^t{(t-\sigma )}^{-2\alpha }{\Vert \psi (\sigma )\Vert }_{L_2(U,L^2(G))}^2\mathrm{d}\sigma \right]}^{\frac{\beta }{2}}\mathrm{d}t\hfill \\ \hfill & ⩽c\mathbb{E}\left[{\int }_0^T{\Vert \psi \Vert }_{L_2(U,L^2(G))}^{\beta }\mathrm{d}t\right].\hfill \end{array}$

This concludes the proof.  □

Consider a sequence of cylindrical Wiener processes $({\mathbf{W}}^n)$ over U (see (9.1.1)) with respect to the filtration ${({\mathcal{F}}_t)}_{t⩾0}$. Assume that $({\mathrm{\Psi }}^n)$ is a sequence of progressively ${({\mathcal{F}}_t)}_{t⩾0}$-measurable processes such that ${\mathrm{\Psi }}^n\in L^2(0,T;L_2(U,L^2(G)))$ $\mathbb{P}$-a.s. Suppose there is a cylindrical ${({\mathcal{F}}_t)}_{t⩾0}$-Wiener process W and $\mathrm{\Psi }\in L^2(0,T;L_2(U,L^2(G)))$, progressively ${({\mathcal{F}}_t)}_{t⩾0}$-measurable, such that

$\begin{array}{cc}\hfill {\mathbf{W}}^n& \to \mathbf{W}\mathit{\text{in}}{C}^0([0,T];U_0),\hfill \\ \hfill {\mathrm{\Psi }}^n& \to \mathrm{\Psi }\mathit{\text{in}}{L}^2(0,T;L_2(U,L^2(G))),\hfill \end{array}$

in probability. Then we have

${\int }_0^{\cdot }{\mathrm{\Psi }}^n\mathrm{d}{\mathbf{W}}^n\to {\int }_0^{\cdot }\mathrm{\Psi }\mathrm{d}\mathbf{W}\mathit{\text{in}}{L}^2(0,T;L^2(G)),$

in probability.

Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space with right-continuous filtration ${({\mathcal{F}}_t)}_{t⩾0}$, let $u_0\in L^2(\mathrm{\Omega },{\mathcal{F}}_0,\mathbb{P};L^2(G))$ and assume that Φ satisfies (9.2.5) and is progressively ${({\mathcal{F}}_t)}_{t⩾0}$-measurable. Then there is a progressively ${({\mathcal{F}}_t)}_{t⩾0}$-measurable process

$u\in L^2(\mathrm{\Omega },\mathcal{F},\mathbb{P};L^2(0,T;W_0^{1,2}(G)))$

such that (9.2.6) holds. We also have $u\in C([0,T];L^2(G))$ a.s. with

$\mathbb{E}[\underset{t\in (0,T)}{\sup }{\int }_G|u(t){|}^2\mathrm{d}x]<\infty .$

The solution u is unique in the above class.

We will split the proof in several steps. First we approximate (9.2.6) by a Galerkin–Ansatz. Then we show a priori estimates and finally we pass to the limit. Note that the uniqueness of solutions is obvious: the difference of two potential solutions solves a deterministic heat equation with zero initial datum.

Step 1: approximation

We will solve (9.2.6) by a finite dimensional approximation. So we need an appropriate basis of $W_0^{1,2}(G)$. A good choice is the set of eigenfunctions of the Laplace operator. There is a smooth orthonormal system ${(w_k)}_{k\in \mathbb{N}}\subset L^2(G)$ and $({\lambda }_k)\in (0,\infty )$ such that

${\int }_G\mathrm{\nabla }{w}_k\cdot \mathrm{\nabla }\phi \mathrm{d}x={\lambda }_k{\int }_G{w}_k\phi \mathrm{d}x\forall \phi \in W_0^{1,2}(G).$

We seek the approximate solution $u_N$ such that

$u_N=\sum _{k=1}^N{c}_N^i{w}_k={\mathbf{C}}_N\cdot {\mathbf{w}}_N,{\mathbf{w}}_N=(w_1,...,w_N),$

where ${\mathbf{C}}_N=(c_N^i):\mathrm{\Omega }\times (0,T)\to {\mathbb{R}}^N$. Therefore, we would like to solve the system

$\begin{array}{cc}\hfill {\int }_G& \mathrm{d}{u}_N{w}_k\mathrm{d}x+{\int }_G\mathrm{\nabla }{u}_k\cdot \mathrm{\nabla }{w}_k\mathrm{d}x\mathrm{d}t={\int }_G\Phi \mathrm{d}{\mathbf{W}}_{\sigma }^N\cdot w_k\mathrm{d}x,k=1,...,N,\hfill \\ \hfill & u_N(0)={\mathcal{P}}_N{u}_0.\hfill \end{array}$

Here ${\mathcal{P}}^N:L_{\mathrm{div}}^2(G)\to {\mathcal{X}}_N:=\mathrm{span}\{w_1,...,w_N\}$ is the orthogonal projection, i.e.

${\mathcal{P}}_{N}u=\sum _{k=1}^N{\langle u,w_k\rangle }_{L^2}{w}_k.$

Equation (9.2.8) is to be understood $\mathbb{P}$-a.s. and for a.e. t and we assume

${\mathbf{W}}^N(\sigma )=\sum _{k=1}^N{\mathbf{e}}_k{\beta }_k(\sigma )={\mathbf{e}}^N\cdot {\mathit{\beta }}^N(\sigma ).$

It is equivalent to solving

$\{\begin{array}{cc}\hfill \mathrm{d}{\mathbf{C}}_N& =\mathbf{\Lambda }{\mathbf{C}}_N\mathrm{d}t+\mathbf{\Sigma }\mathrm{d}{\mathit{\beta }}_t^N,\hfill \\ \hfill {\mathbf{C}}_N(0)& ={\mathbf{C}}_0,\hfill \end{array}$

where $\mathbf{\Lambda },\mathbf{\Sigma }\in {\mathbb{R}}^{N\times N}$ with

${\mathbf{\Lambda }}_{ij}={\delta }_{ij}{\lambda }_j,{\mathbf{\Sigma }}_{ij}={\int }_G\Phi {\mathbf{e}}_i{w}_j\mathrm{d}x.$

As (9.2.9) is a linear system of ODEs we obtain a unique solution (with a.s. continuous trajectories) by the results of Section 8.4.

Step 2: a priori estimates

We apply Itô's formula to the function $f(\mathbf{C})=\frac{1}{2}|\mathbf{C}{|}^2$, thereby obtaining

$\begin{array}{cc}\hfill \frac{1}{2}{\Vert u_N(t)\Vert }_{L^2(G)}^2& =\frac{1}{2}{\Vert {\mathbf{C}}_N(0)\Vert }_{L^2(G)}^2+{\int }_0^t{\mathbf{C}}_N\cdot \mathrm{d}{({\mathbf{C}}_N)}_{\sigma }+\frac{1}{2}{\int }_0^{t}I:\mathrm{d}{\langle \langle {\mathbf{C}}^N\rangle \rangle }_{\sigma }\hfill \\ \hfill & =\frac{1}{2}{\Vert {\mathcal{P}}_N{u}_0\Vert }_{L^2(G)}^2-{\int }_0^t{\int }_G\mathrm{\nabla }{u}_N\cdot \mathrm{\nabla }{u}_N\mathrm{d}x\mathrm{d}\sigma \hfill \\ \hfill & +{\int }_G{\int }_0^t{u}_N\Phi \mathrm{d}{\mathbf{W}}_{\sigma }^N\mathrm{d}x+\frac{1}{2}\sum _k{\int }_0^t{\mathrm{\Sigma }}_{kk}^2\mathrm{d}x\mathrm{d}\sigma .\hfill \end{array}$

In the above we used (9.2.7) and

$\begin{array}{cc}\hfill \mathrm{d}{c}_N^k& =-{\int }_G\mathrm{\nabla }{u}_N\cdot \mathrm{\nabla }{w}_k\mathrm{d}x\mathrm{d}t+{\int }_G\Phi \mathrm{d}{\mathbf{W}}_t^N{w}_k\mathrm{d}x,\hfill \\ \hfill \langle \langle c_N^i,c_N^j\rangle \rangle & =\langle \langle \sum _{k=1}^N{\int }_0^{\cdot }{\int }_G\Phi {\mathbf{e}}_j{w}_k\mathrm{d}x\mathrm{d}{\beta }_k,\sum _{l=1}^N{\int }_0^{\cdot }{\int }_G\Phi {\mathbf{e}}_i{w}_l\mathrm{d}x\mathrm{d}{\beta }_l\rangle \rangle \hfill \\ \hfill & =\sum _{k=1}^N\sum _{l=1}^N\langle \langle {\int }_0^{\cdot }{\int }_G\Phi {\mathbf{e}}_j{w}_k\mathrm{d}x\mathrm{d}{\beta }_k,{\int }_0^{\cdot }{\int }_G\Phi {\mathbf{e}}_i{w}_l\mathrm{d}x\mathrm{d}{\beta }_l\rangle \rangle \hfill \\ \hfill & =\sum _{k=1}^N\left({\int }_G\Phi {\mathbf{e}}_j{w}_k\mathrm{d}x\right)\left({\int }_G\Phi {\mathbf{e}}_i{w}_k\mathrm{d}x\right)={({\mathbf{\Sigma }}^2)}_{ij}.\hfill \end{array}$

Now we obtain, taking the supremum in time and taking expectations, that

$\begin{array}{cc}\hfill & \mathbb{E}[\underset{t\in (0,T)}{\sup }{\int }_G|u_N(t){|}^2\mathrm{d}x+{\int }_0^T{\int }_G|\mathrm{\nabla }{u}_N{|}^2\mathrm{d}x\mathrm{d}\sigma ]\hfill \\ \hfill & ⩽c\mathbb{E}[{\int }_G|u_0{|}^2\mathrm{d}x+\sum _{k=1}^N{\int }_0^T{\mathrm{\Sigma }}_{kk}^2\mathrm{d}t+\underset{t\in (0,T)}{\sup }|J(t)|],\hfill \end{array}$

where we set

$J(t)={\int }_G{\int }_0^t{u}_N\Phi \mathrm{d}{\mathbf{W}}_{\sigma }^N\mathrm{d}x.$

By Hölder's inequality and an account of ${\Vert w_k\Vert }_2=1$, straightforward calculations show

$\begin{array}{cc}\hfill \mathbb{E}\left[\sum _{k=1}^N{\int }_0^T{\mathrm{\Sigma }}_{kk}^2\mathrm{d}x\mathrm{d}t\right]& =\mathbb{E}[\sum _{k=1}^N{\int }_0^T{\int }_G|\Phi {\mathbf{e}}_k{|}^2\mathrm{d}x\mathrm{d}t]\hfill \\ \hfill & ⩽\mathbb{E}[\sum _{k=1}^{\infty }{\int }_0^T{\int }_G|\Phi {\mathbf{e}}_k{|}^2\mathrm{d}x\mathrm{d}t]\hfill \\ \hfill & =\mathbb{E}\left[{\int }_0^T{\Vert \Phi \Vert }_{L_2(U,L^2(G))}^2\mathrm{d}t\right].\hfill \end{array}$

On account of Burkholder–Davis–Gundi inequality (Lemma 9.1.1) and Young's inequality we obtain

$\begin{array}{cc}\hfill \mathbb{E}[\underset{t\in (0,T)}{\sup }|J(t)|]& =\mathbb{E}[\underset{t\in (0,T)}{\sup }|{\int }_0^t{\int }_G{u}_N\Phi \mathrm{d}x\mathrm{d}{\mathbf{W}}_{\sigma }^N\left|\right]\hfill \\ \hfill & =\mathbb{E}[\underset{t\in (0,T)}{\sup }|{\int }_0^t\sum _{k=1}^N{\int }_G{u}_N\Phi {\mathbf{e}}_k\mathrm{d}x\mathrm{d}{\beta }_k(\sigma )\left|\right]\hfill \\ \hfill & ⩽c\mathbb{E}{\left[{\int }_0^T\sum _{k=1}^N{\left({\int }_G{u}_N\Phi {\mathbf{e}}_k\mathrm{d}x\right)}^2\mathrm{d}t\right]}^{\frac{1}{2}}\hfill \\ \hfill & ⩽c\mathbb{E}{\left[\right({\int }_0^T(\sum _{k=1}^N{\int }_G|u_N{|}^2\mathrm{d}x{\int }_G|\Phi {\mathbf{e}}_k{|}^2\mathrm{d}x)\mathrm{d}t]}^{\frac{1}{2}}\hfill \\ \hfill & ⩽c\mathbb{E}{[\underset{t\in (0,T)}{\sup }{\int }_G|u_N{|}^2\mathrm{d}x({\int }_0^T\sum _{k=1}^{\infty }{\int }_G|\Phi {\mathbf{e}}_k{|}^2\mathrm{d}x\mathrm{d}t\left)\right]}^{\frac{1}{2}}\hfill \\ \hfill & ⩽\delta \mathbb{E}[\underset{t\in (0,T)}{\sup }{\int }_G|u_N{|}^2\mathrm{d}x]+c(\delta )\mathbb{E}\left[{\int }_0^T{\Vert \Phi \Vert }_{L_2(U,L^2(G))}^2\mathrm{d}t\right],\hfill \end{array}$

for any arbitrary $\delta >0$. If δ is sufficiently small this finally proves

$\begin{array}{cc}\hfill & \mathbb{E}[\underset{t\in (0,T)}{\sup }{\int }_G|u_N(t){|}^2\mathrm{d}x+{\int }_0^T{\int }_G|\mathrm{\nabla }{u}_N{|}^2\mathrm{d}x\mathrm{d}\sigma ]\hfill \\ \hfill & ⩽c\mathbb{E}[{\int }_G|u_0{|}^2\mathrm{d}x+{\int }_0^T{\Vert \Phi \Vert }_{L_2(U,L^2(G))}^2\mathrm{d}t].\hfill \end{array}$

By our assumptions the right-hand-side is finite.

Step 3: passage to the limit

Due to (9.2.11) and passing to a subsequence we obtain a limit function u:

$u_N\rightharpoonup u\text{in}{L}^2(\mathrm{\Omega },\mathcal{F},\mathbb{P};L^2(0,T;W_0^{1,2}(G))).$

We also have that $u\in L^{\infty }(0,T;L^2(G))$ a.s. and

$\mathbb{E}[\underset{t\in (0,T)}{\sup }{\int }_G|u(t){|}^2\mathrm{d}x]<\infty .$

We compute for $\psi \in L^2(\mathrm{\Omega }\times (0,T))$ and $\phi \in W_0^{1,2}(G)$

$\begin{array}{cc}\hfill & \mathbb{E}[{\int }_0^T{\int }_{G}u(t)\psi (t)\phi \mathrm{d}x\mathrm{d}t]=\underset{N\to \infty }{\lim }\mathbb{E}[{\int }_0^T{\int }_G{u}_N(t)\psi (t)\phi \mathrm{d}x\mathrm{d}t]\hfill \\ \hfill & =\underset{N\to \infty }{\lim }\mathbb{E}[{\int }_0^T{\int }_G{u}_N(t)\psi (t){\mathcal{P}}_N\phi \mathrm{d}x\mathrm{d}t]\hfill \\ \hfill & =\underset{N\to \infty }{\lim }\mathbb{E}[{\int }_0^T({\int }_G{\mathcal{P}}_N{u}_0\psi (t)\phi \mathrm{d}x\mathrm{d}t+{\int }_0^t{\int }_G\mathrm{\nabla }{u}_N\psi (t)\cdot \mathrm{\nabla }\phi \mathrm{d}x\mathrm{d}\sigma \hfill \\ \hfill & +{\int }_G{\int }_0^t\psi {\mathcal{P}}_N\phi \Phi \mathrm{d}{\mathbf{W}}_{\sigma }^N\mathrm{d}x\left)\right].\hfill \end{array}$

We have to pass to the limit in all terms. The first integral converges as ${\mathcal{P}}_{N}w\to w$ in $L^2(G)$ for any $w\in L^2(G)$. The second term converges because of (9.2.12). Finally, we use

${\mathbf{W}}^N\to \mathbf{W}\text{in}{L}^2(\mathrm{\Omega },\mathcal{F},\mathbb{P};C([0,T];U_0)),$

for the stochastic integral (recall Lemma 9.1.4). All together we obtain

$\begin{array}{cc}\hfill & \mathbb{E}[{\int }_0^T{\int }_{G}u(t)\psi (t)\phi \mathrm{d}x\mathrm{d}t]\hfill \\ \hfill & =\mathbb{E}\left[{\int }_0^T{\int }_G(u_0\psi (t)\phi +{\int }_0^t\mathrm{\nabla }\psi (t)\cdot \mathrm{\nabla }\phi \mathrm{d}\sigma +{\int }_0^t\psi \phi \Phi \mathrm{d}{\mathbf{W}}_{\sigma })\mathrm{d}x\right],\hfill \end{array}$

which implies that $\mathbb{P}$-a.s.

${\int }_{G}u(t)\phi \mathrm{d}x={\int }_G{u}_0\phi \mathrm{d}x-{\int }_0^t{\int }_G\mathrm{\nabla }u\cdot \mathrm{\nabla }\phi \mathrm{d}x\mathrm{d}\sigma +{\int }_G\phi \left({\int }_0^t\Phi \mathrm{d}{\mathbf{W}}_{\sigma }\right)\mathrm{d}x$

as ψ was arbitrary.

Step 4: continuity of u

Interpreted as an element of $W^{-1,2}(G)$ we can write $\mathbb{P}$-a.s.

$u(t)={\int }_0^t\mathrm{\Delta }u\mathrm{d}\sigma +{\int }_0^t\Phi \mathrm{d}{\mathbf{W}}_{\sigma }$

for a.e. t. For any $\alpha <\frac{1}{2}$ the deterministic integral belongs $\mathbb{P}$-a.s. to the class

$W^{1,2}(0,T;W^{-1,2}(G))\subset C^{\alpha }([0,T];W^{-1,2}(G)).$

For the stochastic integral we have

${\int }_0^{\cdot }\Phi \mathrm{d}{\mathbf{W}}_{\sigma }\in C([0,T];{(L^2(G))}^{\prime })\subset C([0,T];W^{-1,2}(G))\mathbb{P}\text{-a.s.}$

This follows from the construction and our assumption (9.2.5). Combining both facts shows that $u\in C([0,T];W^{-1,2}(G))$. This and (9.2.11) yields

$u\in C_w([0,T];L^2(G))\mathbb{P}\text{-a.s.}$

We want to strengthen (9.2.14) and obtain continuity with respect to the norm topology. We apply Itô's formula in infinite dimensions, Theorem 9.1.39, with $\mathcal{H}=L^2(G)$, $\mathcal{V}=W_0^{1,2}(G)$, $X=u$, $Y{=}_{\mathcal{V}{}^{\prime }}{\langle \mathrm{\nabla }u,\mathrm{\nabla }\cdot \rangle }_{\mathcal{V}}$ and $M={\int }_0^{\cdot }\Phi \mathrm{d}{\mathbf{W}}_{\sigma }$. We have

$\begin{array}{cc}\hfill {\int }_G|u(t){|}^2\mathrm{d}x& ={\int }_G|u(0){|}^2\mathrm{d}x-{\int }_0^t{\int }_G|\mathrm{\nabla }u{|}^2\mathrm{d}x\mathrm{d}\sigma \hfill \\ \hfill & +2{\int }_G{\int }_0^{t}u\Phi \mathrm{d}{\mathbf{W}}_{\sigma }\mathrm{d}x+{\int }_0^t{\Vert \Phi \Vert }_{L_2(U,L^2(G))}^2\mathrm{d}t.\hfill \end{array}$

As the right-hand-side is continuous so is the left-hand-side, i.e.

$[0,T]\ni t\mapsto {\int }_G|u(t){|}^2\mathrm{d}x$

is $\mathbb{P}$-a.s. continuous. This and (9.2.14) implies $u\in C([0,T];L^2(G))$ a.s.  □