Hypocoercivity of Langevin-type dynamics on abstract smooth manifolds

TL;DR

This paper proves exponential convergence to equilibrium for Langevin-type stochastic differential equations on abstract smooth manifolds, extending hypocoercivity techniques to complex geometric settings with explicit convergence rates.

Contribution

It extends hypocoercivity analysis to Langevin dynamics on abstract manifolds, incorporating geometric structures and measure spaces, with explicit convergence rate computations.

Findings

Exponential decay to equilibrium established for Langevin dynamics on manifolds.

Extension of hypocoercivity methods to nonlinear geometric settings.

Explicit convergence rates derived for a broad class of SDEs on manifolds.

Abstract

In this article we investigate hypocoercivity of Langevin-type dynamics in nonlinear smooth geometries. The main result stating exponential decay to an equilibrium state with explicitly computable rate of convergence is rooted in an appealing Hilbert space strategy by Dolbeault, Mouhot and Schmeiser. This strategy was extended in [GS14] to Kolmogorov backward evolution equations in contrast to the dual Fokker-Planck framework. We use this mathematically complete elaboration to investigate wide ranging classes of Langevin-type SDEs in an abstract manifold setting, i.e. (at least) the position variables obey certain smooth side conditions. Such equations occur e.g. as fibre lay-down processes in industrial applications. We contribute the Lagrangian-type formulation of such geometric Langevin dynamics in terms of (semi-)sprays and point to the necessity of fibre bundle measure spaces to specify the model Hilbert space.