When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|X_0\|_H^2=2\int_0^t\langle X_s,\varphi_s\rangle_H+\|\Phi_s\|_{\operatorname{HS}(U_0,\:H)}^2\:{\rm d}s+2\int_0^t\langle X_s,\phi_s\rangle_H\:{\rm d}W_s\tag1,$$ where $U_0:=Q^{\frac12}U$.
Now, my $X$ is the solution of the (at this point formal) infinite-dimensional SDE $${\rm d}X_t=b(t,X_t){\rm d}t+\sigma(t,X_t){\rm d}W_t\tag2,$$ where $$b(t,x):=\nabla\cdot g_1(t,\nabla x)\nabla x\;\;\;\text{for }(t,x)\in[0,T]\times H^2(\Lambda)$$ and $$\sigma(t,x)y:=g_2(t,\nabla x)v\;\;\;\text{for }v\in H\text{ and }(t,u)\in[0,T]\times V.$$ Here $\Lambda\subseteq\mathbb R^d$ is bounded and open, $H:=L^2(\Lambda)$ and $V:=H^1(\Lambda)$.
I know that I most probably would need to adjust the formulation to ensure that a solution can exist at all. But I'm actually only interested in the numerical simulation of $(2)$. However, in my implementation, I need (an estimate of) the quantity $\operatorname E\left[\|X_t\|_H^2\right]$.
Can I somehow analytically compute or at least numerically estimate this quantity? If not, can I analytically compute or numerically estimate a (sharp) upper bound?
If that's helpful, I'd be willing to assume that the $g_i$ are "separable" in the sense $g_i(t,x)=\alpha_i(t)h_i(x)$.
