The flux of particles in a one-dimensional Fleming-Viot process

TL;DR

This paper analyzes a Fleming-Viot particle system on a semi-infinite line with biased diffusion, deriving explicit solutions, studying flux transitions, and predicting large N corrections, revealing parallels with Fisher-KPP dynamics.

Contribution

It provides explicit solutions for the Fleming-Viot process on a semi-infinite line and explores flux transitions and large N corrections, extending understanding of particle systems with absorbing boundaries.

Findings

Explicit solutions for the Fleming-Viot process are obtained.

Flux transitions can be induced by modifying diffusion rules near the origin.

Predictions for large N corrections of absorbed particle flux are derived.

Abstract

The Fleming-Viot process describes a system of $N$ particles diffusing on a graph with an absorbing site. Whenever one of the particles is absorbed, it is replaced by a new particle at the position of one of the $N-1$ remaining particles. Here we consider the case where the particles lie on the semi-infinite line with a biased diffusion towards the origin which is the absorbing site. In the large $N$ limit, the evolution of the density becomes deterministic and has a number of characteristics similar to the Fisher-KPP equation: a one-parameter family of steady state solutions, dependence of the long time asymptotics on the initial conditions, Bramson logarithmic shift, etc. One noticeable difference, however, is that in the Fleming-Viot case, the solution can be computed explicitly for arbitrary initial conditions and at an arbitrary time. By modifying the diffusion rule near the origin, one can produce a transition in the flux of absorbed particles, very similar to the pushed-pulled transition in travelling waves. Lastly, using a cut-off approximation (which is known to be correct in the theory of travelling waves), we derive a number of predictions for the leading large $N$ correction of the flux of absorbed particles.