Weak Poincaré inequalities for convergence rate of degenerate diffusion processes

Grothaus, Martin; Wang, Feng‐Yu

doi:10.1214/18-aop1328

Cited by 29 publications

(66 citation statements)

References 23 publications

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“…This encompasses target distributions which possess subgeometric tail decay, and are typically referred to as 'heavy-tailed'. To do this, we will utilize the approach of [15], where such inequalities were studied for degenerate diffusions. Our main abstract result will be a convergence result of the form (1.1) where the rate function ξ is in fact subgeometric.…”

Section: Introductionmentioning

confidence: 99%

“…

We extend the hypocoercivity framework for piecewise-deterministic Markov process (PDMP) Monte Carlo established in [2] to heavy-tailed target distributions, which exhibit subgeometric rates of convergence to equilibrium. We make use of weak Poincaré inequalities, as developed in the work of [15], the ideas of which we adapt to the PDMPs of interest. On the way we report largely potential-independent approaches to bounding explicitly solutions of the Poisson equation of the Langevin diffusion and its first and second derivatives, required here to control various terms arising in the application of the hypocoercivity result.

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confidence: 99%

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Subgeometric hypocoercivity for piecewise-deterministic Markov process Monte Carlo methods

Andrieu

Dobson

Wang

2021

Electron. J. Probab.

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Subgeometric hypocoercivity for piecewise-deterministic Markov process Monte Carlo methods

Andrieu

Dobson

Wang

2021

Electron. J. Probab.

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“…/ norm was constructed in which the dynamics contracts. Although structurally different but similar in spirit, we refer the reader to the papers [4,11,13] and the references therein for scenarios where Langevin dynamics produces subgeometric rates of convergence either due to absence of friction on certain particles or weak Poincaré inequalities being satisfied.…”

Section: Introductionmentioning

confidence: 99%

Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

Herzog

Mattingly

2019

Comm Pure Appl Math

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We study Langevin dynamics of N particles on ℝd interacting through a singular repulsive potential, e.g., the well‐known Lennard‐Jones type, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof of the main result relies on an explicit construction of a Lyapunov function. In contrast to previous results for such systems, our result implies geometric convergence to equilibrium starting from an essentially optimal family of initial distributions. © 2019 Wiley Periodicals, Inc.

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“…1 and γ > 0 are constants, which is said to be hypocoercive. The literature on this topic is very rich and active, and several elegant techniques have been developed; we mention, without aiming to be exhaustive, generalizations of the Γ-calculus by Baudoin [6] and Monmarché [36], entropy functional techniques by Dolbeault et al [16,17] and Arnold [4], the shrinkage/enlargenment approach by Gualdini et al [25], Bouin et al [10] and Mischler and Mouhot [35], generalized quadratic and Dirichlet form approaches by Ottobre et al [37] and Grothaus and Stilgenbauer [23], respectively, a weak Poincaré inequality approach by Grothaus and Wang [24], a direct spectral approach for some toy models by Gadat and Miclo [22], and a spectral approach combined with techniques from non-harmonic analysis by Patie and Savov [38] and Patie et al [39]. We also mention the fundamental memoir by Villani [45], noting that the techniques developed therein were inspired by the work of Talay [43].…”

Section: Introductionmentioning

confidence: 99%

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

Patie¹,

Vaidyanathan²

2020

Kinetic &Amp; Related Models

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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

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Weak Poincaré inequalities for convergence rate of degenerate diffusion processes

Cited by 29 publications

References 23 publications

Subgeometric hypocoercivity for piecewise-deterministic Markov process Monte Carlo methods

Subgeometric hypocoercivity for piecewise-deterministic Markov process Monte Carlo methods

Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

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

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