On quantitative hypocoercivity estimates based on Harris-type theorems
arXiv:2203.00096 · doi:10.1063/5.0089698
Abstract
This review concerns recent results on the quantitative study of convergence towards the stationary state for spatially inhomogeneous kinetic equations. We focus on analytical results obtained by means of certain probabilistic techniques from the ergodic theory of Markov processes. These techniques are sometimes referred to as Harris-type theorems. They provide constructive proofs for convergence results in the (or total variation) setting for a large class of initial data. The convergence rates can be made explicit (both for geometric and sub-geometric rates) by tracking the constants appearing in the hypotheses. Harris-type theorems are particularly well-adapted for equations exhibiting non-explicit and non-equilibrium steady states since they do not require prior information on the existence of stationary states. This allows for significant improvements of some already-existing results by relaxing assumptions and providing explicit convergence rates. We aim to present Harris-type theorems, providing a guideline on how to apply these techniques to the kinetic equations at hand. We discuss recent quantitative results obtained for kinetic equations in gas theory and mathematical biology, giving some perspectives on potential extensions to nonlinear equations.
arXiv comment40 pages, typos are corrected, new references are added and structure of the paper has changed
References in corpus (22)
- Decay and Continuity of Boltzmann Equation in Bounded Domains
- 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
- Subgeometric ergodicity of strong Markov processes
- Practical drift conditions for subgeometric rates of convergence
- Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions
- On a kinetic FitzHugh-Nagumo model of neuronal network
- Asymptotic Stability of the Boltzmann Equation with Maxwell Boundary Conditions
- A non-conservative Harris ergodic theorem
- Asymptotic behaviour of neuron population models structured by elapsed-time
- Measure solutions to the conservative renewal equation
- On an irreducibility type condition for the ergodicity of nonconservative semigroups
- On the speed of approach to equilibrium for a collisionless gas
- Spectral gap for the growth-fragmentation equation via Harris's Theorem
- Hypocoercivity of linear kinetic equations via Harris's Theorem
- The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states
- Fractional Fokker-Planck Equation with General Confinement Force
- Cercignani-Lampis boundary in the Boltzmann theory
- A multiple time renewal equation for neural assemblies with elapsed time model
- Equilibrating effect of Maxwell-type boundary condition in highly rarefied gas
- Existence of a Non-Equilibrium Steady State for the Non-linear BGK equation on an interval
- On the asymptotic behaviour of a run and tumble equation for bacterial chemotaxis