Title: Selection principle for the Fleming-Viot process with drift $-1$ Show Chinese title

Title: 漂移项Fleming-Viot过程的选择原理 $-1$

Abstract: We consider the Fleming-Viot particle system consisting of $N$ identical particles evolving in $\mathbb{R}_{>0}$ as Brownian motions with constant drift $-1$. Whenever a particle hits $0$, it jumps onto another particle in the interior. It is known that this particle system has a hydrodynamic limit as $N\rightarrow\infty$ given by Brownian motion with drift $-1$ conditioned not to hit $0$. This killed Brownian motion has an infinite family of quasi-stationary distributions (QSDs), with a Yaglom limit given by the unique QSD minimising the survival probability. On the other hand, for fixed $N<\infty$, this particle system converges to a unique stationary distribution as time $t\rightarrow\infty$. We prove the following selection principle: the empirical measure of the $N$-particle stationary distribution converges to the aforedescribed Yaglom limit as $N\rightarrow\infty$. The selection problem for this particular Fleming-Viot process is closely connected to the microscopic selection problem in front propagation, in particular for the $N$-branching Brownian motion. The proof requires neither fine estimates on the particle system nor the use of Lyapunov functions. Show Chinese abstract

Abstract: 我们考虑由$N$个相同粒子组成的Fleming-Viot粒子系统，在$\mathbb{R}_{>0}$中作为具有常数漂移$-1$的布朗运动演化。每当一个粒子碰到$0$时，它会跳到内部的另一个粒子上。已知当$N\rightarrow\infty$时，该粒子系统有一个流体动力学极限，由不碰到$0$的具有漂移$-1$的布朗运动给出。这种被杀死的布朗运动有一个无限族拟平稳分布（QSDs），其Yaglom极限是由最小化生存概率的唯一QSD给出的。 另一方面，对于固定的$N<\infty$，这个粒子系统随时间$t\rightarrow\infty$收敛到唯一的平稳分布。 我们证明了以下选择原则：$N$粒子平稳分布的经验测度随着$N\rightarrow\infty$趋近于前述的 Yaglom 极限。 这个特定 Fleming-Viot 过程的选择问题与前缘传播中的微观选择问题密切相关，特别是对于$N$分支布朗运动。 证明既不需要对粒子系统的精细估计，也不需要使用 Lyapunov 函数。