Stochastic Perturbation of Sweeping Process for Uniformly Prox-Regular Moving Sets

In this paper, we study the existence of solutions to a sweeping process 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 consider several geometric assumptions and establish important relationships between them.

💡 Research Summary

The paper addresses the existence and uniqueness of solutions to stochastic sweeping processes when the moving constraint set varies continuously in time with respect to the Hausdorff distance and takes uniformly prox‑regular values. The authors first introduce a minimal geometric framework for such moving sets, collecting the most common hypotheses found in the literature (denoted H1–H5) and clarifying their logical relationships. In particular, they identify (H5) as the weakest verifiable condition for Skorokhod‑type problems, and they provide practical criteria that apply to constraints defined as finite intersections of sublevel sets.

With this geometric setting in place, the authors study the deterministic Skorokhod (or sweeping) problem under continuous perturbations. They prove new well‑posedness results and obtain quantitative bounds on the total variation of the correction term (the “reflection” or “projection” measure). These bounds are uniform for families of equicontinuous perturbations, which is crucial for later stochastic analysis.

The core contribution concerns the stochastic differential inclusion

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