正半群稳定性的Lyapunov方法：带示例的综述
A Lyapunov approach to stability of positive semigroups: An overview with illustrations
AI总结：
本文综述了正半群稳定性分析的Lyapunov方法，涵盖半群比较与共轭原理，并通过Markov半群、条件随机过程、Langevin扩散和耦合谐振子等实例展示其应用。
摘要：
对可能时变的正半群在非紧状态空间上（包括Neumann和Dirichlet边界条件）的稳定性分析是一个众所周知的困难课题。这些关键问题出现在应用数学的多个领域，包括非线性滤波、稀有事件分析、分支过程、物理学和分子化学。本文概述了一些近期基于Lyapunov的方法，主要聚焦于设计Lyapunov函数的实用且有力的工具。这些技术包括半群比较以及在具有局部Lipschitz边界的非必有界流形上的共轭原理。文中讨论的所有Lyapunov方法均在多种情形下加以说明，范围从一般状态空间上的传统Markov半群到更复杂的条件随机过程，后者可能被限制在某些非必有界区域上，包括局部Lipschitz和光滑超曲面边界、Langevin扩散以及耦合谐振子。
英文摘要：
The stability analysis of possibly time varying positive semigroups on non necessarily compact state spaces, including Neumann and Dirichlet boundary conditions is a notoriously difficult subject. These crucial questions arise in a variety of areas of applied mathematics, including nonlinear filtering, rare event analysis, branching processes, physics and molecular chemistry. This article presents an overview of some recent Lyapunov-based approaches, focusing principally on practical and powerful tools for designing Lyapunov functions. These techniques include semigroup comparisons as well as conjugacy principles on non necessarily bounded manifolds with locally Lipschitz boundaries. All the Lyapunov methodologies discussed in the article are illustrated in a variety of situations, ranging from conventional Markov semigroups on general state spaces to more sophisticated conditional stochastic processes possibly restricted to some non necessarily bounded domains, including locally Lipschitz and smooth hypersurface boundaries, Langevin diffusions as well as coupled harmonic oscillators.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
