Asymptotic behaviour of coupled random dynamical systems with multiscale aspects

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.