| Contents |
| The aim of this talk is to apply an extended version of the modern, powerful and simple
abstract Hilbert space strategy for proving hypocoercivity that has been developed originally
by Dolbeault, Mouhot and Schmeiser. It is well-known that hypocoercivity methods imply an
exponential decay to equilibrium with explicit computable rate of convergence for degenerate
evolution equations. In the stated extension we introduced important domain issues that have
not been considered before. Necessary conditions for proving hypocoercivity need then only
to be verified on a fixed operator core of the evolution operator. Additionally, the setting is
suitably reformulated to incorporate also strongly continuous semigroups solving the Kolmo-
gorov equation as an abstract Cauchy problem. In this way it can be applied to the Langevin
dynamics arising in Statistical Mechanics and Mathematical Physics. In this application, the
strongly continuous contraction semigroup can be constructed via using Kato-perturbation
tools. Moreover, via using techniques from the theory of Dirichlet forms, it admits a natural
representation as the transition kernel of diffusion process solving the underlying equation in
the martingale sense. Summarizing, we provide the first complete elaboration of the Hilbert
space hypocoecivity theorem for the degenerate Langevin dynamics in this hypocoercivity
setting. |
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