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arXiv · 2601.15411

Asymptotic behaviour of coupled random dynamical systems with multiscale aspects

D. Russell Luke·Johannes-Carl Schnebel·Mathias Staudigl·Juan Peypouquet·Siqi Qu
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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.

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D. Russell Luke, Johannes-Carl Schnebel, Mathias Staudigl, Juan Peypouquet, Siqi Qu. 2026-07-27. Asymptotic behaviour of coupled random dynamical systems with multiscale aspects. https://arxiv.org/abs/2601.15411

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