论文标题
基于HARRIS型定理的定量估计值
On quantitative hypocoercivity estimates based on Harris-type theorems
论文作者
论文摘要
这篇综述涉及有关在空间不均匀动力学方程的固定状态的定量研究的最新结果。我们专注于通过Markov过程的Ergodic理论通过某些概率技术获得的分析结果。这些技术有时称为Harris型定理。它们为收敛提供了建设性的证据，从而为大量初始数据提供了$ l^1 $（或总变化）设置。可以通过跟踪出现在假设中的常数来显式（对于几何和子几何速率）。 Harris型定理特别适应表现出非阐释和非平衡稳态的方程式，因为它们不需要事先有关固定状态存在的信息。这可以通过放松假设并提供明确的收敛率来显着改善一些已经存在的结果。我们的目标是介绍Harris型定理，提供有关如何将这些技术应用于手头动力学方程式的指南。我们讨论了气体理论和数学生物学中动力学方程获得的最新定量结果，对非线性方程的潜在扩展提供了一些观点。
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 $L^1$ (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.
