Let's say we have a (parabolic) Cauchy problem:

$$ (1) \hspace{0.5cm} u_t (x,t)+A(u) \cdot u_x (x,t))=\nu \cdot u_{xx} (x,t) + \epsilon \cdot f(u) \cdot W, $$ $$(2) \hspace{0.5cm} u(x,0)=u_0(x), $$

where $x \in L \subseteq \mathbb{R}^d, d \geq 1$, $t \in [0,T]$, $u \in \mathbb{R}^n, n \geq 1$ and $W$ is white noise stochastic process. And let's suppose that the problem $(1)-(2)$ has some solution.

My question is: What conditions need to be satisfied in order to pass to the limit $\nu \rightarrow 0$ or $\epsilon \rightarrow 0$ or both?

Remark: The problem $(1)-(2)$ is just a special case of the problem

$$ (3) \hspace{0.5cm} u_t (x,t)+A(u) \cdot u_x (x,t))=\sum_{i=1}^{m}\nu_i \cdot g_i + \sum_{i=1}^{k}\epsilon_i \cdot s_i,$$ $$(4) \hspace{0.5cm} u(x,0)=u_0(x), $$

where the first term(s) on the right hand side of the $(3)$ represents some deterministic term(s) (which can depend on the derivatives on $u$ as in the $(1)$), and the second term(s) represents some stochastic process (not necessarily the white noise process).

So what would be conditions that would able us to pass in any/some/all of the limits in $(3)$ or in $(1)$?

In order to make things simple you could take:

d=1(means that the space variable is one dimensional)

n=1(means that we have an equation instead of the system)

Examples or/and some theoretical answers would be great. And of course if anyone knows some good reference in the literature that studies this type of problems, let me know.

Help with this is really appreciated.


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