Sweeping Processes Perturbed by Rough Signals

TL;DR

This work advances the theory of sweeping processes by treating perturbations driven by rough signals in the $p$-variation range $p\in[1,3[$. It establishes pathwise existence for both Young ($p\in[1,2[$) and rough ($p\in[2,3[$) regimes, under a moving convex constraint with interior, via a Skorokhod-type decomposition and compactness arguments. A near-monotonicity condition near the normal cone yields uniqueness at $p=1$, and an explicit discrete approximation scheme is shown to converge to the unique solution in this and related settings; the results extend to pathwise perturbations directed by a fractional Brownian motion with $H>1/3$ through rough-path methods. Collectively, the paper provides a comprehensive pathwise framework for perturbed sweeping processes, with concrete existence, uniqueness criteria, approximation schemes, and stochastic-noise extensions relevant to variational and Skorokhod-type problems.

Abstract

This paper deals with the existence, the uniqueness and an approximation scheme of the solution to sweeping processes perturbed by a continuous signal of finite $p$-variation with $p\in [1,3[$. It covers pathwise stochastic noises directed by a fractional Brownian motion of Hurst parameter greater than $1/3$.

Theorems & Definitions (44)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 2.4
• Proposition 2.5
• proof
• Definition 2.6
• Proposition 2.7
• Proposition 2.8
• Proposition 2.9