Configurational transition in a Fleming - Viot-type model and probabilistic interpretation of Laplacian eigenfunctions

Abstract

We analyze and simulate a two dimensional Brownian multi-type particle system with death and branching (birth) depending on the position of particles of different types. The system is confined in the two dimensional box, whose boundaries act as the sink of Brownian particles. The branching rate matches the death rate so that the total number of particles is kept constant. In the case of m types of particles in the rectangular box of size a, b and elongated shape a ≫ b we observe that the stationary distribution of particles corresponds to the m-th Laplacian eigenfunction. For smaller elongations a > b we find a configurational transition to a new limiting distribution. The ratio a/b for which the transition occurs is related to the value of the m-th eigenvalue of the Laplacian with rectangular boundaries.

Fleming–Viot selects the minimal quasi-stationary distribution: The Galton–Watson case

“…and show that there exist positive constants C 1 and C 2 such that 11) for all N, see Proposition 8.1 later. The issue here is a uniform bound for the correlations of the empirical distribution of Fleming-Viot at sites x, y ∈ N at fixed time t. This was carried out in [1].…”

“…The first term in the right hand side represents the displacement of mass due to the jumps of the process and the second term represents the mass going from each x to 0 and then coming instantaneously to y. In 1996, Burdzy, Holyst, Ingerman and March [11] introduced a genetic particle system called Fleming-Viot named after models proposed in [18], which can be seen as a particle system mimicking the evolution (1.1). The particle system can be built from a process with absorption Z t called driving process; the position Z t is interpreted as a genetic trait, or fitness, of an individual at time t. In the N-particle Fleming-Viot system, each trait follows independent dynamics with the same law as Z t except when one of them hits state 0, a lethal trait: at this moment the individual adopts the trait of one of the other individuals chosen uniformly at random.…”

Tagged Particle Limit for a Fleming-Viot Type System

“…Let M N (dx) be the unique stationary distribution of the process {x N (·)} (the measure exists according to [3]) and Φ 1 (x) be the first eigenfunction of the Laplacian with Dirichlet boundary conditions normalized such that it integrates to one over D. The next corollary is based on [2] and [11]. We are now ready to state the main results of this paper, theorems 2 and 3 and their corollaries.…”

“…are the jump times of the counting process A N 1 (·) defined before Proposition 6. Here we define m(T ) such that τ m(T ) ≤ T < τ m(T )+1 , which is possible a.s., since for finite N there are finitely many jumps in a finite time interval with probability one (proven in [2]). To explain the last inequality we notice that the jumps present in (3.16) are zero if the partition is taken to be exactly l = m(T ) + 1 and t i = τ i , 0 ≤ i ≤ l − 1, plus the last endpoint τ l = T , a construction we always can make unless max…”

“…As soon as one of the particles reaches the boundary ∂D, an independent Brownian particle is created at one of the sites of the survivors, chosen with uniform probability. The model (in lattice and continuous version) is due to Burdzy, Ho lyst, Ingerman and March in [2], and later studied (in its present continuous version) by some of the same authors in [3], where a law of large numbers is established for the empirical measures at fixed times. The present paper continues the investigation from [11] where a hydrodynamic limit for the joint law of the empirical measure and the average number of redistributions (boundary hits) was proven.…”