标题： $L^2$-具有$α$-稳定Lévy噪声的随机哈密顿系统的指数遍历性 显示英文标题

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

摘要： 基于Villani的次协方差方法\cite{Villani}，Dolbeault、Mouhot和Schmeiser\cite{DMS}建立了一个新的简单框架，以直接研究动能Fokker-Planck方程解的$L^2$-指数收敛到平衡态。如今，在\cite{DMS}中提出的通用框架被称为DMS次协方差框架。随后，Grothaus和Stilgenbauer\cite{Grothaus}在动能Fokker-Planck设置中构建了DMS框架的对偶版本。无论在\cite{DMS}中的抽象DMS框架及其在\cite{Grothaus}中的对偶版本，所涉及的稠密定义线性算子都被假定分解为两部分，其中一部分是对称的，另一部分是反对称的。 因此，现有的DMS框架不适用于研究带有$\alpha$稳定Lévy噪声的随机Hamilton系统中的$L^2$指数遍历性，其中相关无穷小生成元的一部分是反对称的，而另一部分则不是对称的。 在本文中，我们将开发一种分数动能Fokker-Planck设置下的DMS框架的对偶版本，其中所考虑的稠密定义线性算子的一部分不需要是对称的。 作为直接应用，我们探讨了带有$\alpha$稳定Lévy噪声的随机Hamilton系统的$L^2$指数遍历性。 证明还基于非局部稳定类似Dirichlet形式的Poincaré不等式和分数Riesz位势的位势理论。 显示英文摘要

摘要： 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 $\alpha$-stable L\'{e}vy 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 $\alpha$-stable L\'evy noises. The proof is also based on Poincar\'e inequalities for non-local stable-like Dirichlet forms and the potential theory for fractional Riesz potentials.