Asymptotic behaviour of coupled random dynamical systems with multiscale aspects

TL;DR

This paper investigates the long-term behavior of stochastic differential inclusions with multiscale effects, providing convergence rates, large deviations results, and connections to penalty algorithms in constrained optimization.

Contribution

It introduces a novel analysis of multiscale stochastic systems with constraints, establishing asymptotic convergence, finite-time rates, and links to discrete penalty algorithms.

Findings

Proves convergence of the stochastic dynamics under a Legendre transform condition.

Establishes exponential concentration of trajectories around the solution set.

Connects continuous-time dynamics to penalty-regulated algorithms via discretization.

Abstract

We examine a class of stochastic differential inclusions involving multiscale effects designed to solve a class of generalized variational inequalities. This class of problems contains constrained convex non-smooth optimization problems, constrained saddle-point problems and various equilibrium problems in economics and engineering. In order to respect constraints we adopt a penalty approach, introducing an explicit time-dependency into the evolution system. The resulting dynamics are described in terms of a non-autonomous stochastic evolution equation governed by maximally monotone operators in the drift and perturbed by a Brownian motion. We study the asymptotic behavior, as well as finite time convergence rates in terms of gap functions. The condition we use to prove convergence involves a Legendre transform of the function describing the set C, a condition first used by Attouch and Czarnecki (J. Differ. Equations, Vol. 248, Issue 6, 2010) in the context of deterministic evolution equations. We also establish a large deviations principle showing that individual trajectories exhibit exponential concentration around the solution set. Finally we show how our continuous-time approach relates to penalty-regulated algorithms of forward-backward type after performing a suitable Euler-Maruyama discretisation.