论文标题
McKean-Vlasov动力学的Fleming-Viot过程
The Fleming-Viot Process with McKean-Vlasov Dynamics
论文作者
论文摘要
Fleming-Viot粒子系统由$ n $相同的粒子组成，在域中扩散$ u \ u \ subset \ mathbb {r}^d $。每当粒子击中边界$ \ partial u $时，该粒子就会跳到内部的另一个粒子上。众所周知，该系统为准平台分布（QSD）提供了粒子表示，并且在其域边界处被杀死的给定扩散的生存条件。我们将这些结果扩展到McKean-Vlasov动力学的情况。我们证明，根据相应的Fleming-Viot粒子系统的流体动力学极限，可以获得以给定的McKean-Vlasov过程的生存条件。然后，我们表明，如果目标将McKean-Vlasov流程杀死为QSD，则以$ t \ rightArrow \ infty $的价格收敛，则可以从相应的$ n $ particle fleming-Viot系统的固定分布中获得这样的QSD，为$ n \ rightarrow \ rightarrow \ rightarrow \ inftty $。
The Fleming-Viot particle system consists of $N$ identical particles diffusing in a domain $U \subset \mathbb{R}^d$. Whenever a particle hits the boundary $\partial U$, that particle jumps onto another particle in the interior. It is known that this system provides a particle representation for both the Quasi-Stationary Distribution (QSD) and the distribution conditioned on survival for a given diffusion killed at the boundary of its domain. We extend these results to the case of McKean-Vlasov dynamics. We prove that the law conditioned on survival of a given McKean-Vlasov process killed on the boundary of its domain may be obtained from the hydrodynamic limit of the corresponding Fleming-Viot particle system. We then show that if the target killed McKean-Vlasov process converges to a QSD as $t \rightarrow \infty$, such a QSD may be obtained from the stationary distributions of the corresponding $N$-particle Fleming-Viot system as $N\rightarrow\infty$.
