A lot of problems in PDE theory are solved in the following way: The original problem is quite hard and we can't solve it, so we make the approximation problem that we can solve. Than we go back and with the new informations we got, we solve the original problem. In this question I am interested in the "stochastic approximations". Let me explained it in the example.
Let's say that we have a PDE problem:
$$(1) \hspace{1cm} \begin{cases} u_t+f(u)_x=0 \\[2ex] u(x,0)=u_0 (x) \end{cases} $$
Here we have system of conservation laws. Of course $(1)$ could be any other PDE problem (I am just using $(1)$ to ilustrate a point).
Let's say our "approximation problem" is given with:
$$(2) \hspace{1cm} \begin{cases} u_t+f(u)_x=\epsilon \cdot g(u)W(t) \\[2ex] u(x,0)=u_0 (x) \end{cases} $$
Here u $\in \mathbb{R}^n$, W(t) is a white noise, $\epsilon$ is a parameter (that will go to zero eventually).
Another type of the "approximation problem" could be
$$(3) \hspace{1cm} \begin{cases} u_t+f(u)_x=0 \\[2ex] u(x,0)=u_0^W (x) \end{cases} $$
where $u_0^W$ is some random variable.
In the one more aproximation we could make flux into a stochastic variable. I won't discuss here if the $(2)$ or $(3)$ are easier to solve than $(1)$.
My question(s) is this: If we assume that we know the solution of the stochastic problem, how would we show the convergence of the solution of $(2)$ or $(3)$ to the solution of $(1)$. For example, in the problem $(2)$, when $\epsilon\rightarrow 0$, how would we show that the solution of (2) always converge to the solution of (1)?
Additionaly,what could be the other stochastic problems (besides $(2)$ and $(3)$ and stochastic flux) that we could use to approximate the original problem?
If anyone know some good references in the literature that deal with this kind of problems write it down.
