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

TL;DR

This paper investigates the convergence of stochastic integrals driven by Wiener processes, providing new results that facilitate analysis of SPDEs with weak temporal convergence and applying these to stochastic transport and conservation laws.

Contribution

It offers novel convergence results for stochastic integrals with weakly converging integrands, especially useful for SPDEs with singular behavior, supported by stability results for related equations.

Findings

Established convergence results for stochastic integrals with weak temporal convergence.

Provided stability results for stochastic transport equations.

Applied findings to 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.