My question concerns solutions to the nonlinear stochastic heat equation:
$$\begin{align}u_{t}&=u_{xx}+u^{\gamma}\dot{F}, && \gamma\geq 1,&& t>0& 0\leq x\leq J,\\ u(t,0)&=u(t,J)=0 \end{align}$$
where $\dot{F}=\dot{F}(t,x)$ is spatially coloured noise such that $Cov(\dot{F}(t,x),\dot{F}(s,y))=\delta(t-s)f(x-y)$. It is given that $u(0,x)$ is non-negative and hence $u$ remains non-negative.
After playing around with Green's functions, I wound up with the following while attempting to find a quadratic variation
$$N(t,x)=\int_{0}^{t}\int_{0}^{J}G(t-s,x,y) L^{\gamma}F(dy,ds)$$ where $L,\gamma$ are constants and $G(t,x,y)$ is the fundamental solution of the heat equation on $[0,J]$: $$\begin{align} G_{t}(t,x,y)&=G_{xx}(t,x,y)\\ G(t,0,y)&=G(t,J,y)=0\\ G(0,x,y)&=\delta(x-y). \end{align}$$ I wish to compute $E[N^{2}(t,x)]$ but I find myself at a loss by conflicting notation across the papers I have read. Were this standard white noise, I would simply apply Ito's isometry, but the introduction of $f(x-y)$ to provide spatially coloured noise has me somewhat confused. I don't believe Ito's isometry can be directly applied. Any assistance interpreting what $E[N^{2}(t,x)]$ looks like would be greatly appreciated.