Infinite Systems of Competing Brownian Particles

Andrey Sarantsev

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arXiv · 1403.4229

Infinite Systems of Competing Brownian Particles

Andrey Sarantsev

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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).

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BibTeX RIS

Andrey Sarantsev. 2014-03-17. Infinite Systems of Competing Brownian Particles. https://arxiv.org/abs/1403.4229

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