Hypocoercivity for degenerate Kolmogorov equations and applications to the Langevin dynamics

Abstract

We present an extension of the powerful Hilbert space hypocoercivity method that was developed originally by Dolbeault, Mouhot and Schmeiser. We focus attention on including important domain issues that have not been considered before. The setting can be used to provide a complete elaboration of the hypocoercivity theorem for the degenerate Langevin dynamics.In an interesting research article Dolbeault, Mouhot and Schmeiser developed a simple strategy for proving exponential decay to zero of specific strongly continuous semigroups with associated degenerate (e.g. Fokker-Planck) generator L in a Hilbert space framework, see [2, Sec. 1.3]. The strategy may be called hypocoercivity, cf. [6]. It was the great idea of Dolbeault, Mouhot and Schmeiser to find a suitable entropy functional, which is equivalent to the underlying Hilbert space norm, for measuring the exponential convergence to equilibrium, see [2, Sec. 1.3]. Neglecting any domain issues and questions concerning the construction of the semigroup first, the conditions in the framework of Dolbeault, Mouhot and Schmeiser can then rather easily be checked in applications and imply the desired ergodicity behavior of the semigroup.However, in the method from [2, Sec. 1.3] domain issues are not included, thus calculations are established algebraically only. In this sense, the framework is not yet complete. Our extension now concerns Kolmogorov type evolution equations which are again formulated as an abstract Cauchy problem in a Hilbert space framework. We include the domain issues and extend the setting from [2] to the case in which we know an operator core for the Kolmogorov generator L of the strongly continuous semigroup (abbreviated C 0 -semigroup) of interest. Conditions (H1)-(H4) from [2, Sec. 1.3] are adapted in a suitable way and it suffices to verify them on the fixed operator core of L only. Together with some data conditions (D), this simplifies domain issues and is suitable for showing essential m-dissipativity of L and essential selfadjointness of a related operator occurring in the framework. As an application, we prove hypocoercivity of the degenerate Langevin dynamics.The underlying proceeding summarizes some of the main results obtained in our articles [3] and [4] (or see [5, Ch. 2]). We do not present any proofs. For more details, the reader is referred to these references. The hypocoercivity resultBelow we state the main result. First, we introduce the necessary assumptions.