HYPOCOERCIVITY FOR LINEAR KINETIC EQUATIONS CONSERVING MASS

By Jean Dolbeault, Clément Mouhot, Christian Schmeiser. 2010

354 citations recorded by the source corpus.

We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models like the linear Boltzmann equation or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed.

Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4a10e663b999befd4ffb9fb0c68dda4ce21ce3ae

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In Applied Mathematics

In Atomic and Molecular Physics, and Optics

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Citations in the corpus

Posterior Computation with the Gibbs Zig-Zag Sampler (2020, methodology): “he lines of Bierkens et al. (2018b); Deligiannidis et al. (2018). Moreover, the derivation of -dependent spectral estimates for the generator of the GZZ process using the hypocoercivity framework of Dolbeault et al. (2015) (see Andrieu et al., 2018 for an adoption of that framework to PDMPs) would b”
Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation (2014, background): “(2) We did not include it in the statement for the sake of clarity but our method of proof can recover the semigroup decay estimate in L(2)(μ) as a consequence of the known decay estimate in H(1)(μ) proved in [14, 16, 24], which provides an alternative argument to those in [8, 9].”
Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows (2015, background): “[17,18,25,26] and the references therein or the text [47] for an overview of the basic ideas and a general study of hypocoercivity applied to linear operators in Hörmander form.”
Underdamped Langevin MCMC: A non-asymptotic analysis (2017, background): “The continuous-time process has been studied extensively [12, 13, 14, 15, 16, 17, 18, 19, 20].”
Factorization for non-symmetric operators and exponential H-theorem (2010, methodology): “Let us first introduce the notion of hypodissipative operators and discuss its relation with the classical notions of dissipative operators and coercive operators as well as its relation with the recent terminology of hypocoercive operators (see mainly [112] and then [89, 62, 44] for related referen”
Couplings and quantitative contraction rates for Langevin dynamics (2017, background): “Several analytic approaches to convergence to equilibrium for kinetic Fokker-Planck equations have been proposed during the last 15 years [10, 17, 27, 25, 26, 40, 42, 11, 7, 33, 19, 20, 3].”