Singular perturbations for a subelliptic operator^★,★★

¹ Dipartimento di Matematica “Tullio Levi-Civita”, Università degli Studi di Padova, Via Trieste 63, 35121, Padova, Italy
² Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Padova, Via Gradenigo 6/b, 35131, Padova, Italy
³ IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

^* Corresponding author: claudio.marchi@unipd.it

Received: 23 February 2017
Accepted: 8 September 2017

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 ϵ → 0, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and terminal data are explicitly given in terms of the invariant measure for the ergodic operator.