Hypocoercivity for Kolmogorov backward evolution equations and applications

Abstract

In this article we extend the modern, powerful and simple abstract Hilbert space strategy for proving hypocoercivity that has been developed originally by Dolbeault, Mouhot and Schmeiser. As well-known, hypocoercivity methods imply an exponential decay to equilibrium with explicit computable rate of convergence. Our extension is now made for studying the long-time behavior of some strongly continuous semigroup generated by a (degenerate) Kolmogorov backward operator L. Additionally, we introduce several domain issues into the framework. Necessary conditions for proving hypocoercivity need then only to be verified on some fixed operator core of L. Furthermore, the setting is also suitable for covering existence and construction problems as required in many applications. The methods are applicable to various, different, Kolmogorov backward evolution problems. As a main part, we apply the extended framework to the (degenerate) spherical velocity Langevin equation. The latter can be seen as some kind of an analogue to the classical Langevin equation in case spherical velocities are required. This model is of important industrial relevance and describes the fiber lay-down in the production process of nonwovens. For the construction of the strongly continuous contraction semigroup we make use of modern hypoellipticity tools and pertubation theory.

References in corpus (7)

• Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus
• Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation
• Asymptotic analysis for the generalized langevin equation
• Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes
• Geometric Langevin equations on submanifolds and applications to the stochastic melt-spinning process of nonwovens and biology
• Geometry, mixing properties and hypocoercivity of a degenerate diffusion arising in technical textile industry
• Geometric Ergodicity of a Hypoelliptic Diffusion Modelling The Melt-Spinning Process of Nonwoven Materials

Cited by in corpus (16)

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• Hilbert space hypocoercivity for the Langevin dynamics revisited
• Couplings and quantitative contraction rates for Langevin dynamics
• Subgeometric hypocoercivity for piecewise-deterministic Markov process Monte Carlo methods
• Hypercontractivity for Stochastic Hamiltonian Systems
• Self-repelling diffusions on a Riemannian manifold
• Ergodicity for Neutral Type SDEs with Infinite Length of Memory
• Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise
• Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces
• Weak Poincaré Inequalities for Convergence Rate of Degenerate Diffusion Processes
• Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations
• Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature
• A simple hypocoercivity analysis for the effective Mori-Zwanzig equation
• Closability of Quadratic Forms Associated to Invariant Probability Measures of SPDEs