Hypocoercivity for linear kinetic equations conserving mass

Abstract

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 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.

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

“…Using the framework of [15], we conjecture that the assumption on the boundedness of the Hessian of V can be substantially weakened and more quantitative decay estimates (in particular with respect to μ and ν) can be obtained. This approach has recently been successfully applied to equilibrium and nonequilibirum Langevin dynamics, see [27,53].…”

“…Remark 12 In [14], the authors consider the task of finding optimal perturbations J for the nonreversible overdamped Langevin dynamics given in (15). In the Gaussian case this optimization problem turns out be equivalent to the one considered in this section.…”

Hypocoercivity for geometric Langevin equations motivated by fibre lay‐down models arising in industrial application

“…The method was developed in Dolbeault et al [12,13] algebraically in the sense that for example, domain issues of involved operators are not considered there. Note that we basically just have to check these conditions in applications, since the machinery works in a quite abstract Hilbert space setting.…”

“…In a nutshell, we use an elliptic a priori estimate -namely the estimate in Grothaus and Stilgenbauer [7,Proposition A.6] which can be proven likewise to Dolbeault et al [13,Proposition 5] This estimate requires all three conditions (C1)-(C3). In fact, we need the full set (C) of conditions on the potential to prove the technical boundedness condition (H4).…”

Trend to equilibrium and spectral localization properties for the linear Boltzmann equation

“…Subsequently, Hérau and Nier [14], Hérau [13], Mouhot and Neumann [17], Villani [22], Dolbeault, Mouhot and Schmeiser [8], among others, proposed several methods to study various kinetic models (in particular a potential V = 0 is allowed).…”