Convergent finite difference schemes for stochastic transport equations

TL;DR

This paper develops convergent finite difference schemes for stochastic transport equations with low-regularity velocities, demonstrating stability and convergence under relaxed conditions compared to deterministic cases.

Contribution

It introduces novel difference schemes with proven stability and convergence for stochastic transport equations with less restrictive assumptions.

Findings

Established $L^2$ stability for the schemes

Proved convergence under weaker conditions than deterministic cases

Utilized discrete duality and backward parabolic schemes for analysis

Abstract

We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish $L^2$ stability and convergence of the difference approximations under conditions that are less strict than those required for deterministic transport equations. The $L^2$ estimate, crucial for the analysis, is obtained through a discrete duality argument and a comprehensive examination of a class of backward parabolic difference schemes.