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

Convergence of stochastic integrals driven by Wiener processes Wₙ, with Wₙ → W almost surely in Cₜ, is crucial in analyzing SPDEs. Our focus is on the convergence of the form ∫₀T Vₙ\, d Wₙ → ∫₀T V\, d W, where ₙ\ is bounded in Lᵖ(Ω × [0,T];X) for a Banach space X and some finite $p > 2$. This is challenging when Vₙ 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 L¹ time translation estimate on Vₙ, an estimate that is easily verified in many SPDEs. However, this estimate alone does not guarantee strong compactness of (ω,t)↦ Vₙ(ω,t). Our findings, especially pertinent to equations exhibiting singular behavior, are substantiated by establishing several stability results for stochastic transport equations and conservation laws.