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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.07668 |
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| _version_ | 1866913197044269056 |
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| author | Jianhai, Bao Jian, Wang |
| author_facet | Jianhai, Bao Jian, Wang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07668 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^2$-exponential ergodicity of stochastic Hamiltonian systems with $α$-stable Lévy noises Jianhai, Bao Jian, Wang Probability 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. |
| title | $L^2$-exponential ergodicity of stochastic Hamiltonian systems with $α$-stable Lévy noises |
| topic | Probability |
| url | https://arxiv.org/abs/2401.07668 |