Stochastic Perturbation of Sweeping Processes Driven by Continuous Uniformly Prox-Regular Moving Sets

TL;DR

This paper establishes the existence and uniqueness of solutions for stochastic sweeping processes with uniformly prox-regular moving sets that vary continuously, broadening the theoretical framework for stochastic differential equations in time-dependent domains.

Contribution

It introduces a minimal geometric framework for such sets, clarifies hypotheses relations, and provides practical conditions for constraints as intersections of sublevel sets.

Findings

Proved existence of weak and strong solutions.

Established pathwise uniqueness for the stochastic differential equations.

Applied results to constraints defined as intersections of sublevel sets.

Abstract

In this paper, we study the existence of solutions to sweeping processes in the presence of stochastic perturbations, where the moving set takes uniformly prox-regular values and varies continuously with respect to the Hausdorff distance, without smoothness assumptions. We propose a minimal geometric framework for such moving sets, make precise the logical implications between several standard hypotheses in the literature, and provide practical sufficient conditions that apply in particular to constraints defined as finite intersections of sublevel sets. Within this setting, we establish existence of weak and strong solutions and prove pathwise uniqueness for the associated stochastic differential equations reflected in time-dependent domains.