d{2\left( 1-\theta \right) },\infty
\right) ;t}.$ Moreover we have the following duality relation: $$\label{dual2}\int_0^t\int_{{\cal O}}u\left( s,x\right) v\left( s,x\right)
dxds\le \left\| u\right\| _{\theta;t}\left\| v\right\|_{\theta;t}^*,$$ for any $u\in L_{\theta;t}$ and $v\in L_{\theta;t}^*$ and the following inequality: $$\label{controltheta}
\left\|u\right\|_{\theta;t}\leq
c_1\left(\left\|u\right\|^2_{2,\infty;t}+\left\|\nabla
u\right\|^2_{2,2;t}\right)^{1/2}.$$

Hypotheses
----------

We consider a sequence $((B^i(t))_{t\geq0})_{i\in\mathbb{N}^*}$ of independent Brownian motions defined on a standard filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq0},P)$ satisfying the usual conditions.

Let $A$ be a symmetric second order differential operator defined on the open bounded subset $ {\ensuremath{\mathcal O}}\subset {\mathbb{R}}^d$, with domain $\mathcal{D}(A)$, given by $$A:=-L=-\sum_{i,j=1}^d\partial_i(a^{i,j}\partial_j).$$ We assume that $a=(a^{i,j})_{i,j}$ is a measurable symmetric matrix defined on $\mathcal{O}$ which satisfies the uniform ellipticity condition$$\lambda|\xi|^2\leq\sum_{i,j=1}^d
a^{i,j}(x)\xi^i\xi^j\leq\Lambda|\xi|^2,\ \forall x\in\mathcal{O},\
\xi\in {\mathbb{R}}^d,$$where $\lambda$ and $\Lambda$ are positive constants. The energy associated with the matrix $a$ will be denoted by $$\label{energy}
 \mathcal{E} \left( w,v\right)=\sum_{i,j=1}^d
\int_{{\ensuremath{\mathcal O}}}a^{i,j}(x)\partial_i w(x)\partial_j v(x)\, dx .$$ It’s defined for functions $w,\, v \in  H^1_{0}
(\mathcal{O} )$, or for $w \in H^1_{loc} (\mathcal{O} )$ and $v \in
H^1_{0} (\mathcal{O })$ with compact support.

We assume that we have predictable random functions$$\begin{aligned}
&&f:{\mathbb{R}}_+\times\Omega\times\mathcal{O}\times
{\mathbb{R}}\times {\mathbb{R}}^d\rightarrow
{\mathbb{R}},\\&&g=(g_1,...,g_d):{\mathbb{R}}_+\times\Omega\times\mathcal{O}\times
{\mathbb{R}}\times {\mathbb{R}}^d\rightarrow
{\mathbb{R}}^d,\\&&h=(h_1,...,h_i,...):{\mathbb{R}}_+\times\Omega\times\mathcal{O}\times
{\mathbb{R}}\times {\mathbb{R}}^d\rightarrow {\mathbb{R}}^{\mathbb{N}^*}.\end{aligned}$$ We define $$\begin{split}
 &f ( \cdot
,\cdot,\cdot, 0,0):=f^0, \  g( \cdot,\cdot,\cdot ,0,0) :=g^0 =
(g_1^0,...,g_d^0) \ \mbox{and}
\  h( \cdot,\cdot,\cdot 