分布依赖随机微分方程在平衡点附近的局部收敛性
Local Convergence near Equilibria for Distribution-Dependent SDEs in a Generalized Kantorovich--Rubinstein Metric
AI总结：
本文通过线性化非线性 Markov 半群，将分布依赖 SDE 解在平稳分布附近的局部指数收敛归结为线性算子半群的收敛，并给出稳定性判据与颗粒介质方程实例。
摘要：
由于会呈现相变现象，我们研究分布依赖随机微分方程在平稳分布附近的局部收敛性。通过将与该分布依赖方程相关的非线性 Markov 半群在平稳分布附近线性化，解的局部指数收敛性与一个线性算子半群的指数收敛性联系起来。我们的结果可用作平稳分布局部指数稳定性的判据。文中给出了具体例子，包括具有双势阱景观和二次相互作用的颗粒介质方程，以阐明主要结果。
英文摘要：
Owing to exhibiting phase transitions, we investigate the local convergence near a stationary distribution for distribution dependent stochastic differential equations. By linearizing the nonlinear Markov semigroup associated with the distribution dependent equation around the stationary distribution, the local exponential convergence of the solution is related to the exponential convergence of a semigroup of linear operators. The generation and regularity of the linearized semigroup are investigated, and the Poincaré inequality for the stationary distribution is adapted to derive the exponential convergence of the linearized semigroup. Our results can be used as a criterion for the locally exponential stability of stationary distributions. Concrete examples, including the granular media equation with double-wells landscapes and quadratic interaction, are given to illustrate our main results.
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。
