L^2-exponential ergodicity of stochastic Hamiltonian systems with α-stable Lévy noises
arxiv（2024）
摘要
Based on the hypocoercivity approach due to Villani ,
Dolbeault, Mouhot and Schmeiser 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 is named as the DMS framework for
hypocoercivity. Subsequently, Grothaus and Stilgenbauer builded
a dual version of the DMS framework in the kinetic Fokker-Planck setting. No
matter what the abstract DMS framework in and the dual counterpart
in , 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.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树，研究论文发展脉络
Chat Paper
正在生成论文摘要