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$L^2$-exponential ergodicity of stochastic Hamiltonian systems with $α$-stable Lévy noises

Bao Jianhai, Wang Jian

TL;DR

This work addresses the challenge of proving $L^2$-exponential ergodicity for stochastic Hamiltonian systems driven by α-stable Lévy noises, where the generator lacks a straightforward symmetric decomposition. It introduces a dual DMS hypocoercivity framework that does not require a symmetric part and relies on a nonlocal, stable-like Dirichlet form for the velocity variable. Under assumptions on the potentials $U$ and radial drift $Φ$ ensuring Poincaré inequalities for the marginals and a suitable nonlocal friction term, it proves exponential convergence in $L^2(μ)$ with explicit rates and constants. A concrete corollary with $Φ(v)=\tfrac{1}{2}(d+β)\log(1+|v|^2)$ yields $L^2$-ergodicity for $β\in[α,2α)$, highlighting the practical regime where the method applies.

Abstract

Based on the hypocoercivity approach due to Villani \cite{Villani}, Dolbeault, Mouhot and Schmeiser \cite{DMS} established a new and simple framework to investigate directly the $L^2$-exponential convergence to the equilibrium for the solution to the kinetic Fokker-Planck equation. Nowadays, the general framework advanced in \cite{DMS} is named as the DMS framework for hypocoercivity. Subsequently, Grothaus and Stilgenbauer \cite{Grothaus} builded a dual version of the DMS framework in the kinetic Fokker-Planck setting. No matter what the abstract DMS framework in \cite{DMS} and the dual counterpart in \cite{Grothaus}, the densely defined linear operator involved is assumed to be decomposed into two parts, where one part is symmetric and the other part is anti-symmetric. Thus, the existing DMS framework is not applicable to investigate the $L^2$-exponential ergodicity for stochastic Hamiltonian systems with $α$-stable Lévy noises, where one part of the associated infinitesimal generators is anti-symmetric whereas the other part is not symmetric. In this paper, we shall develop a dual version of the DMS framework in the fractional kinetic Fokker-Planck setup, where one part of the densely defined linear operator under consideration need not to be symmetric. As a direct application, we explore the $L^2$-exponential ergodicity of stochastic Hamiltonian systems with $α$-stable Lévy noises. The proof is also based on Poincaré inequalities for non-local stable-like Dirichlet forms and the potential theory for fractional Riesz potentials.

$L^2$-exponential ergodicity of stochastic Hamiltonian systems with $α$-stable Lévy noises

TL;DR

This work addresses the challenge of proving -exponential ergodicity for stochastic Hamiltonian systems driven by α-stable Lévy noises, where the generator lacks a straightforward symmetric decomposition. It introduces a dual DMS hypocoercivity framework that does not require a symmetric part and relies on a nonlocal, stable-like Dirichlet form for the velocity variable. Under assumptions on the potentials and radial drift ensuring Poincaré inequalities for the marginals and a suitable nonlocal friction term, it proves exponential convergence in with explicit rates and constants. A concrete corollary with yields -ergodicity for , highlighting the practical regime where the method applies.

Abstract

Based on the hypocoercivity approach due to Villani \cite{Villani}, Dolbeault, Mouhot and Schmeiser \cite{DMS} established a new and simple framework to investigate directly the -exponential convergence to the equilibrium for the solution to the kinetic Fokker-Planck equation. Nowadays, the general framework advanced in \cite{DMS} is named as the DMS framework for hypocoercivity. Subsequently, Grothaus and Stilgenbauer \cite{Grothaus} builded a dual version of the DMS framework in the kinetic Fokker-Planck setting. No matter what the abstract DMS framework in \cite{DMS} and the dual counterpart in \cite{Grothaus}, the densely defined linear operator involved is assumed to be decomposed into two parts, where one part is symmetric and the other part is anti-symmetric. Thus, the existing DMS framework is not applicable to investigate the -exponential ergodicity for stochastic Hamiltonian systems with -stable Lévy noises, where one part of the associated infinitesimal generators is anti-symmetric whereas the other part is not symmetric. In this paper, we shall develop a dual version of the DMS framework in the fractional kinetic Fokker-Planck setup, where one part of the densely defined linear operator under consideration need not to be symmetric. As a direct application, we explore the -exponential ergodicity of stochastic Hamiltonian systems with -stable Lévy noises. The proof is also based on Poincaré inequalities for non-local stable-like Dirichlet forms and the potential theory for fractional Riesz potentials.
Paper Structure (6 sections, 12 theorems, 150 equations)

This paper contains 6 sections, 12 theorems, 150 equations.

Table of Contents

  1. Introduction and Main Result
  2. Background
  3. Setting
  4. Main result
  5. A General DMS framework
  6. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.2

Assume that $d>2-\alpha$, and that both $($${\bf A}_U$$)$ and $($${\bf A}_\Phi$$)$ are satisfied. Then, the process $(X_t,V_t)_{t\ge0}$ solving E1 is $L^2$-exponentially ergodic, i.e., there exist constants $c,\lambda>0$ such that for all $f\in L^2(\mu)$ and $t>0$, where $(P_t)_{t\ge0}$ is the Markov semigroup generated by $(X_t,V_t)_{t\ge0}$ and $\mu$, defined in E1-, is an invariant probability

Theorems & Definitions (28)

  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 18 more