In the study of conservation laws in partial differential equations relatively often we see this two problems (problem (1) more than problem (2)):
Deterministic Cauchy problem: $$(1) \hspace{1cm} \begin{cases} u_t+f(u)_x=0 \\[2ex] u(x,0)=u_0 (x) \end{cases} $$
Stochastic Cauchy problem: $$(2) \hspace{1cm} \begin{cases} u_t+f(u)_x=\lambda \cdot g(u)W(t) \\[2ex] u(x,0)=u_0 (x) \end{cases} $$
Here u $\in \mathbb{R}^n$, W(t) is a stochastic process such as white noise (but it could be any other), $\lambda$ is a constant parameter and $u_0 (x)$ is initial data.
My question is: When $\lambda\rightarrow 0$, does solution of (2) always converge to the solution of (1)? Intuitively to me that makes sense but showing that analytically looks quit hard. In stochastic problems we work on a given probability space so maybe the convergence is in distribution (or some other kind other than pointwise/almost sure convergence)? Also if we think of the problem (2) as the approximation problem for a problem (1), we would need some compactness in order to let $\lambda\rightarrow 0$. But showing that compactness looks very complicated.
If anyone could suggest me some references in the literature connected with this two questions that would be great. And if somone knows the answer and writes it down, that would be pretty good too.
