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

Juan Guillermo Garrido, Nabil Kazi-Tani, Emilio Vilches

TL;DR

The paper addresses stochastic sweeping processes where the moving constraint set $C(t)$ is closed, $\rho$-uniformly prox-regular, and varies continuously in the Hausdorff metric, without smoothness assumptions. It develops a minimal geometric framework and shows that the deterministic sweeping process is well-posed with BV-type a priori bounds, then extends the analysis to stochastic perturbations, proving existence (weak and strong) and pathwise uniqueness under broad conditions. The results rely on a measure-theoretic formulation of sweeping dynamics, a Skorokhod-type interpretation, and a careful study of geometric hypotheses and their relationships, culminating in strong existence results via a Yamada–Watanabe approach. This work broadens the applicability of sweeping process theory to time-dependent, nonconvex, nonsmooth moving sets with stochastic perturbations, without requiring boundedness or smooth boundary assumptions, and provides quantitative BV bounds for perturbations.

Abstract

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.

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

TL;DR

The paper addresses stochastic sweeping processes where the moving constraint set is closed, -uniformly prox-regular, and varies continuously in the Hausdorff metric, without smoothness assumptions. It develops a minimal geometric framework and shows that the deterministic sweeping process is well-posed with BV-type a priori bounds, then extends the analysis to stochastic perturbations, proving existence (weak and strong) and pathwise uniqueness under broad conditions. The results rely on a measure-theoretic formulation of sweeping dynamics, a Skorokhod-type interpretation, and a careful study of geometric hypotheses and their relationships, culminating in strong existence results via a Yamada–Watanabe approach. This work broadens the applicability of sweeping process theory to time-dependent, nonconvex, nonsmooth moving sets with stochastic perturbations, without requiring boundedness or smooth boundary assumptions, and provides quantitative BV bounds for perturbations.

Abstract

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.
Paper Structure (13 sections, 23 theorems, 125 equations, 1 figure)

This paper contains 13 sections, 23 theorems, 125 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. Main Geometric Hypotheses
  5. Relationships among the Geometric Assumptions
  6. Deterministic Sweeping Process with Continuous Moving Sets
  7. Stochastic Perturbation of Sweeping Process
  8. Technical Proofs
  9. Proof of Lemma \ref{['C(t)H-statement']}
  10. Proof of Lemma \ref{['lemma_partition']}
  11. Proof of Proposition \ref{['h1h2-bd']}
  12. Proof of Lemma \ref{['upper-semi-support']}
  13. Proof of Proposition \ref{['prop-sublevel-h4-h5']}

Key Result

Proposition 2.2

Let $S\subset \mathbb{R}^d$ be a closed set and $\rho\in ]0,+\infty]$. The following assertions are equivalent:

Figures (1)

  • Figure 1: A uniformly prox-regular set that does not satisfy neither \ref{['H4mgh']} nor \ref{['H5mgh']}.

Theorems & Definitions (52)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Lemma 2.7
  • Proposition 3.1
  • ...and 42 more