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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 $\endgroup$ Dec 8, 2022 at 11:20

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This type of existence of first fixing an omega and then obtaining a solution is pretty standard eg. see "Regularization by noise and flows of solutions for a stochastic heat equation".

Is there some particular reason you need matching subsequences, which seems unlikely if starting with random initial data too?

For some recent results in stochastic pdes with multiplicative noise see: A Wong-Zakai theorem for stochastic PDEs

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  • $\begingroup$ I see a downvote, any feedback is welcome. $\endgroup$ Apr 8, 2023 at 17:59

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