Space-time white noise on PDE

Question

Suppose I want to solve the following PDE \begin{equation} \frac{\partial u}{\partial z} + \alpha \frac{\partial u}{\partial t} = f(z,t) u+g(z,t)\int_{0}^{t}e^{\beta(s-t)}\xi(z,s)ds \end{equation} where $\xi(z,t)$ is a white noise process in space and time obeying the following properties \begin{equation} \mathbb{E}\left[\xi(z,t) \right] = 0;\quad \mathbb{E}\left[\xi(z,t)\xi(z',t') \right] = \delta(z-z')\delta(t-t'), \end{equation} and $\alpha$ and $\beta$ are arbitrary constants. The function $u(z,t)$ is complex-valued and obeys the following conditions \begin{equation} u(z,0) = 0;\quad u(L,t) =u_0(t), \end{equation} We can obtain the expectation of $u$ by using \begin{equation} \left(\frac{\partial}{\partial z} + \alpha\frac{\partial}{\partial t}\right)\mathbb{E}[u] = f(z,t)\mathbb{E}[u] \end{equation} since the integral on the right vanishes when you apply the expectation operator. However, I am more interested in the variance of $u(z,t)$, which requires me to calculate $\mathbb{E}[u^2]$. We may attempt a solution by multiplying both sides by $u$, and using the chain rule on the left to obtain \begin{equation} \frac{1}{2}\left(\frac{\partial}{\partial z} + \alpha \frac{\partial }{\partial t}\right)\mathbb{E}[u^2] = f(z,t)\mathbb{E}[u^2]+g(z,t)\mathbb{E}\left[u\int_{0}^{t}e^{\beta(s-t)}\xi(z,s)ds\right], \end{equation} But now I run into the problem with the expectation on the RHS. Both $u$ and $\xi(z,t)$ are stochastic processes, so to find $\mathbb{E}\left[u\int_{0}^{t}e^{\beta(s-t)}\xi(z,s)ds\right]$ we would require an extra equation. Is there any other way to obtain $\mathbb{E}[u^2]$?