具有硬杀灭的Fleming-Viot粒子系统的时间一致弱收敛
Uniform in time weak convergence for a Fleming-Viot particle system with hard killing
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中文总结 AI 辅助
本文针对Fleming-Viot粒子系统，通过弱混沌传播和屏障函数，给出了经验测度收敛到归一化Dirichlet热方程解的时间一致定量结果。
中文摘要 AI 辅助
本文关注由Burdzy、Hołyst和March（2000）引入的Fleming-Viot粒子系统。在该模型中，$N$个布朗粒子在有界区域$D$内独立演化，直到其中一个粒子到达边界。随后，触及边界的粒子立即跳跃到其他粒子中均匀随机选取的一个粒子的位置。Burdzy、Hołyst和March证明了当$N$趋于无穷时，系统的经验测度收敛到$D$上具有Dirichlet边界条件的热方程的解，该解被重新归一化以具有总质量1。我们的主要结果是该结果的尖锐的、时间一致的、定量的版本。我们采用弱混沌传播的方法，这需要对与$N$粒子系统相关的（后向）Kolmogorov方程以及由Dirichlet热方程的归一化解给出其特征的无穷维输运方程进行仔细研究。证明完全是解析的，大部分技术工作致力于构建屏障函数，用于控制当大多数粒子接近边界时系统的奇异行为。
英文摘要
This paper is concerned with the Fleming-Viot particle system introduced by Burdzy, Hołyst and March (2000). In this model, $N$ Brownian particles evolve independently in a bounded domain $D$ until one of the particles reaches the boundary. Then, the particle which has hit the boundary instantaneously jumps to the location of one of the other particles, chosen uniformly at random. Burdzy, Hołyst and March showed that when $N$ tends to infinity, the empirical measure of the system converges to a solution to the heat equation on $D$ with Dirichlet boundary conditions, renormalized to have total mass $1$. Our main result is a sharp, uniform-in-time, quantitative version of this result. We employ the method of weak propagation of chaos, which necessitates a careful study of the (backward) Kolmogorov equations associated to the $N$-particle systems, and the infinite-dimensional transport equation whose characteristics are given by renormalized solutions of the Dirichlet heat equation. The proofs are entirely analytical, and most of the technical effort is devoted to building barrier functions which are used to control the singular behavior of the system when most of the particles approach the boundary.
发表机构
- Université Paris Dauphine-PSL（巴黎第九大学-巴黎文理研究大学）
- Università degli Studi di Padova（帕多瓦大学）
- The University of Chicago（芝加哥大学）
机构由 AI 辅助整理，请以论文原文为准。
