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For given initial data $u_0\in H^k$ for some $k$, I want to prove the existence of solution to some PDE with multiplicative white noise. I modify the SPDE by regularizing it and then use the compactness method(finding a subsequence) to obtain the solution. What I want to prove is that for $a.e.\omega\in\Omega$, there is a $$u(\omega,t,x)\in C([0,T]; H^k)\ for\ some\ T>0.$$

Let us consider the pathwise solution. I mean, the stochastic basis is given in advance.

Even though for fixed $\omega\in\Omega$, I have found one subsequence of the approximate solution which is strong convergence in the space and time variables, i.e, strong convergence in $x$ and $t$, I can not obtain that for $a.e.\omega\in\Omega$, there is a solution because for different $\omega\in\Omega$, the convergent subsequence is different.

So, what condition should be verified to obtain that for $a.e.\omega\in\Omega$, there is a solution $u(\omega,t,x)$?

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  • $\begingroup$ A cheap comment (but it depends strongly on the PDE you're looking): a possible way out is to establish uniqueness of the solution for each of your random events $\omega$ ; then the whole sequence converges $\endgroup$
    – Ayman Moussa
    Commented Dec 8, 2022 at 11:20

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