Based on the hypocoercivity approach due to Villani, Dolbeault, Mouhot and Schmeiser (TAMS’15) 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 due to  Dolbeault, Mouhot and Schmeiser (TAMS’15) is named as the DMS framework for hypocoercivity. Subsequently, Grothaus (JFA’14) builded a dual version of the DMS framework in the kinetic Fokker-Planck setting. No matter what the abstract DMS framework  and the dual counterpart due to Grothaus (JFA’14), 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 stable Levy 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 stable Levy noises. The proof is also based on Poincare inequalities for non-local stable-like Dirichlet forms and the potential theory for fractional Riesz potentials.

11.9.pdf