动力学Fokker-Planck方程的加权$L^2$界与弱耗散性
Weighted $L^2$ bounds for kinetic Fokker-Planck equations and hypocoercivity
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中文总结 AI 辅助
该研究针对带幂律约束势和重尾局部平衡态的动力学Fokker-Planck方程，采用DMS方法与Foster-Lyapunov条件法建立加权范数全局估计，推导熵-熵产生不等式及平衡态收敛速率。
中文摘要 AI 辅助
我们研究具有幂律约束势和重尾局部平衡态的动力学Fokker-Planck方程解的长时间行为。在不依赖先验矩界或微扰 regime 的情况下，我们采用两种方法建立加权范数的全局估计（主要结果）：加权范数与熵耗散的耦合（DMS方法）以及Foster-Lyapunov条件方法。由此，我们建立了熵-熵产生不等式，并针对空间约束增长指数和局部平衡态尾衰减指数确定了收敛到平衡态的速率。为简化起见，我们假设稳态是可分解的。
英文摘要
We study the long-time behaviour of solutions to kinetic Fokker-Planck equations with power law confinement potentials and local equilibria with fat tails. Without relying on \emph{a priori} moment bounds or perturbative regimes, we establish global estimates on weighted norms (main result) using two methods: a coupling of the weighted norms with entropy dissipation (DMS method) and a Foster-Lyapunov condition approach. As a consequence, we establish an entropy - entropy production inequality and the convergence rates to equilibrium in terms of the exponents associated to the growth of the spatial confinement and the tail decay of local equilibria. For sake of simplicity, we assume that stationary states are factorized.
