Asymptotic behaviour for operators of Grushin type: invariant measure and singular perturbations

TL;DR

This paper studies singular perturbation problems involving Grushin-type operators, proving the existence of invariant measures and regular solutions, and demonstrating convergence of value functions to an effective problem.

Contribution

It establishes the existence of invariant measures and regular solutions for Grushin-type operators in singular perturbation problems, leading to convergence results.

Findings

Invariant measure exists for the fast variable dynamics.

Viscosity solutions with logarithmic growth are obtained.

Value functions converge to an effective problem solution.

Abstract

This paper concerns singular perturbation problems where the dynamics of the fast variable evolve in the whole space according to an operator whose infinitesimal generator is formed by a Grushin type second order part and a Ornstein-Uhlenbeck first order part. We prove that the dynamics of the fast variables admits an invariant measure and that the associated ergodic problem has a viscosity solution which is also regular and with logarithmic growth at infinity. These properties play a crucial role in the main theorem which establishes that the value functions of the starting perturbation problems converge to the solution of an effective problem whose operator and initial datum are given in terms of the associated invariant measure.