Convergent finite difference schemes for stochastic transport equations

TL;DR

This work develops convergent finite difference schemes for stochastic transport equations with gradient noise and low-regularity velocity fields. By integrating a discrete Holmgren-type duality framework with a discrete Green's function/parametrix analysis, the authors obtain robust $L^2$ stability under $\nabla\cdot V \in L^p$ with $p>d$, circumventing the need for $L^{\infty}$ bounds typical of deterministic theory. They establish well-posedness and convergence to the unique weak $L^2$ solution, supported by a detailed analysis of constant- and variable-coefficient discrete parabolic problems and their duals. The results highlight a regularising effect of transport noise that enables stable numerical approximation under weaker assumptions, and they provide a rigorous pathway to accurate simulations of stochastic transport phenomena in rough media.

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.

Theorems & Definitions (51)

• Definition 2.1: Weak solution
• Theorem 2.2: Well-posedness
• Remark 2.3
• Lemma 2.4: Itô-to-Stratonovich conversion
• proof
• Lemma 3.1
• Lemma 3.2: Discrete calculus
• Definition 3.3: Solution to the difference scheme
• Theorem 3.4: Main theorem
• Lemma 4.1: Properties of the fundamental solution