A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators

Pierre Patie; Aditya Vaidyanathan

doi:10.3934/krm.2020016

Article Contents

2020, Volume 13, Issue 3: 479-506. Doi: 10.3934/krm.2020016

This issue Previous Article High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials Next Article Numerical study of Bose–Einstein condensation in the Kaniadakis–Quarati model for bosons

A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators

Pierre Patie^1, and
Aditya Vaidyanathan^2, ,

1.
School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, 14853, USA
2.
Center for Applied Mathematics, Cornell University, Ithaca, NY, 14853, USA

^* Corresponding author: Aditya Vaidyanathan
^* Corresponding author: Aditya Vaidyanathan

Received: May 2019

Revised: November 2019

Published: March 2020

This work was partially supported by NSF Grant DMS-1406599 and a CNRS grant

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Abstract

The aim of this paper is to offer an original and comprehensive spectral theoretical approach to the study of convergence to equilibrium, and in particular of the hypocoercivity phenomenon, for contraction semigroups in Hilbert spaces. Our approach rests on a commutation relationship for linear operators known as intertwining, and we utilize this identity to transfer spectral information from a known, reference semigroup $ \widetilde{{P}} = (e^{-t\widetilde{{\mathbf{A}}}})_{t \geqslant 0} $ to a target semigroup $ P $ which is the object of study. This allows us to obtain conditions under which $ P $ satisfies a hypocoercive estimate with exponential decay rate given by the spectral gap of $ \widetilde{{\mathbf{A}}} $. Along the way we also develop a functional calculus involving the non-self-adjoint resolution of identity induced by the intertwining relations. We apply these results in a general Hilbert space setting to two cases: degenerate, hypoelliptic Ornstein-Uhlenbeck semigroups on $ \mathbb{R}^d $, and non-local Jacobi semigroups on $ [0,1]^d $, which have been introduced and studied for $ d = 1 $ in [12]. In both cases we obtain hypocoercive estimates and are able to explicitly identify the hypocoercive constants.

Keywords:

Hypocoercivity,
Spectral theory,
intertwining,
degenerate hypoelliptic Ornstein-Uhlenbeck operators,
integro-differential operators.

Mathematics Subject Classification: Primary: 58J51, 47G20; Secondary: 60J75.

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