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

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 P = (e −t A ) t 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 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 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.spectral gap γ 1 , and we are able to show, under some conditions, that P satisfies a hypocoercive estimate with exponential rate γ 1 , the spectral gap of the reference operator A. As applications of these results we obtain hypocoercive estimates for degenerate, hypoelliptic Ornstein-Uhlenbeck semigroups on R d , and for non-local Jacobi semigroups on [0, 1] d , recently introduced and studied for d = 1 in [12]. In both cases we make explicit the two hypocoercive constants in terms of the initial data.This paper is organized as follows. In the remainder of this section we consider a motivating example and some preliminaries. In Section 2 we state our main results in a general Hilbert space setting and in Section 3 we present our application of these general results to degenerate, hypoelliptic Ornstein-Uhlenbeck semigroups and non-local Jacobi semigroups. Finally, in Section 4 we provide the proofs.1.1. A motivating example. We present a motivating example from [38], which served as an inspiration for this work. Denote by P = (e −tG ) t 0 and P = (e −t G ) t 0 the generalized and classical Laguerre semigroup, which are contraction semigroups on the spaces L 2 (ν) and L 2 (ε), respectively, where ε(x) = e −x , x > 0, and ν is the unique invariant probability density on (0, ∞) for P , see [38, Theorem 1.6(2)]. The operator −G acts on suitable f via

On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity

“…Similar results have been obtained by Patie and Savov [41] as well as Achleitner et al [1]. Our hypocoercive estimate is obtained via intertwining, which suggests that hypocoercivity may be studied purely from this viewpoint, an idea that is further investigated in the recent work [43] by the second and fourth author.…”

Cutoff Thermalization for Ornstein–Uhlenbeck Systems with Small Lévy Noise in the Wasserstein Distance

“…Therefore state-of-the-art analytic methods, at best, are capable of studying the spectrum of A ε (numerically), which yield an upper bound for exponential convergence to the equilibrium μ ε for sufficiently large time in the case of the spectrum lying in the left open complex half-plane. See for instance [88] Section "Hypoelliptic Ornstein-Uhlenbeck semigroups" or Theorem 3.1 in [10]. However, these types of results can only establish (qualitative) upper bounds, which do not reflect the real convergence of p ε t to the equilibrium distribution μ ε .…”

Discrete self-similar and ergodic Markov chains

“…Note that in general, the hypocoercive constants may be difficult to identify and may have little to do with the spectrum. Results in the spirit of ( 1) have already been obtained by Achleitner et al [1], Patie and Savov [29] as well as in Patie and Vaidyanathan [31] where a general framework based on intertwining relation is developed.…”