Infinite Systems of Competing Brownian Particles
Abstract
Consider a system of infinitely many Brownian particles on the real line. At any moment, these particles can be ranked from the bottom upward. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its current rank. The gaps between consecutive particles form the (infinite-dimensional) gap process. We find a stationary distribution for the gap process. We also show that if the initial value of the gap process is stochastically larger than this stationary distribution, this process converges back to this distribution as time goes to infinity. This continues the work by Pal and Pitman (2008). Also, this includes infinite systems with asymmetric collisions, similar to the finite ones from Karatzas, Pal and Shkolnikov (2016).
Keywords
Cite
@article{arxiv.1403.4229,
title = {Infinite Systems of Competing Brownian Particles},
author = {Andrey Sarantsev},
journal= {arXiv preprint arXiv:1403.4229},
year = {2016}
}
Comments
37 pages. Keywords: Reflected Brownian motion, competing Brownian particles, asymmetric collisions, interacting particle systems, weak convergence, stochastic comparison, triple collisions, stationary distribution