Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise

TL;DR

This paper extends hypocoercivity results for Langevin dynamics to cases with multiplicative noise, providing explicit convergence rates and establishing essential m-dissipativity of the associated operators.

Contribution

It introduces a novel analysis of Langevin dynamics with multiplicative noise, proving essential self-adjointness and m-dissipativity for the Kolmogorov operator in this setting.

Findings

Explicit convergence rates depending on diffusion coefficients and potential.

Extension of hypocoercivity theory to weakly differentiable diffusion matrices.

Validation of the stochastic representation via generalized Dirichlet forms.

Abstract

We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behaviour of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker-Planck framework, and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and R\"ockner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.