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

Paola Mannucci; Claudio Marchi; Nicoletta Tchou

doi:10.3934/dcdss.2019008

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2019, Volume 12, Issue 1: 119-128. Doi: 10.3934/dcdss.2019008

This issue Previous Article Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces Next Article

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

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.
Université Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Received: February 2017

Revised: October 2017

Published: July 2018

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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.

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Mathematics Subject Classification: 35B25, 49L25, 35J70, 35H20, 35B37, 93E20.

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References

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