Singular perturbations for a subelliptic operator

TL;DR

This paper investigates singular perturbation problems involving subelliptic operators, proving convergence of value functions to an effective problem with explicit operator and data, and establishing regularity and growth properties of solutions.

Contribution

It introduces a novel analysis of ergodic problems with subelliptic operators, demonstrating convergence and regularity results for singular perturbations.

Findings

Existence of globally Lipschitz solutions for the ergodic problem.

Logarithmic growth bound at infinity for solutions.

Convergence of value functions to an effective problem as perturbation parameter vanishes.

Abstract

We study some classes of singular perturbation problems where the dynamics of the fast variables evolve in the whole space obeying to an infinitesimal operator which is subelliptic and ergodic. We prove that the corresponding ergodic problem admits a solution which is globally Lipschitz continuous and it has at most a logarithmic growth at infinity. The main result of this paper establishes that as $\epsilon\to 0$, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and data are explicitly given in terms of the invariant measure for the ergodic operator.