Convergence of stochastic integrals with applications to transport equations and conservation laws with noise

Abstract

Convergence of stochastic integrals driven by Wiener processes $W_n$, with $W_n\to W$ almost surely in $C_t$, is crucial in analyzing SPDEs. Our focus is on the convergence of the form ${\int }_0^T{V}_n\mathrm{d}{W}_n\to {\int }_0^{T}V\mathrm{d}W$, where $\{V_n\}$ is bounded in $L^p(\Omega \times [0,T];X)$ for a Banach space $X$ and some finite $p>2$. This is challenging when $V_n$ converges to $V$ weakly in the temporal variable. We supply convergence results to handle stochastic integral limits when strong temporal convergence is lacking. A key tool is a uniform mean $L^1$ time translation estimate on $V_n$, an estimate that is easily verified in many SPDEs. However, this estimate alone does not guarantee strong compactness of $(\omega ,t)\mapsto V_n(\omega ,t)$. Our findings, especially pertinent to equations exhibiting singular behavior, are substantiated by establishing several stability results for stochastic transport equations and conservation laws.

31 pages