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 (the most famous example of this technique I know is the Vanishing viscosity method).

A typical problem looks like this:

$$(1) \hspace{1cm} \begin{cases} u_t+f(u)_x=0 \\[2ex] u(x,0)=u_0 (x) \end{cases} $$

Here we have system of conservation laws (given in divergent form).

Let's say our "approximation problem" looks like this:

$$(2) \hspace{1cm} \begin{cases} u_t+f(u)_x=\epsilon \cdot g(u)W(t) \\[2ex] u(x,0)=u_0 (x) \end{cases} $$

(this can be solved). Here u $\in \mathbb{R}^n$, W(t) is a white noise and $\epsilon$ is a parameter that will go to zero eventually.

I have two questions:

  1. When $\epsilon\rightarrow 0$, how would we show that the solution of (2) always go to the solution of (1)? We would need to show some compactness, so that when $\epsilon\rightarrow 0$, we know that limit exists.

  2. More generally, what are the other problems that show up when we try to approximate deterministic problems with stochastic ones? In (2) we added stochastic source, but we also could add randomness in initial conditions or in the flux, for example.

If anyone know some good references in the literature that deal with this kind of problems share it with the rest of us.


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