Stationary Gap Distributions for Infinite Systems of Competing Brownian Particles

Andrey Sarantsev; Li-Cheng Tsai

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arXiv · 1608.00628

Stationary Gap Distributions for Infinite Systems of Competing Brownian Particles

Andrey Sarantsev·Li-Cheng Tsai

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Abstract

Consider the infinite Atlas model: a semi-infinite collection of particles driven by independent standard Brownian motions with zero drifts, except for the bottom-ranked particle which receives unit drift. We derive a continuum one-parameter family of product-of-exponentials stationary gap distributions, with exponentially growing density at infinity. This result shows that there are infinitely many stationary gap distributions for the Atlas model, and hence resolves a conjecture of Pal and Pitman (2008) in the negative. This result is further generalized for infinite systems of competing Brownian particles with generic rank-based drifts.

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BibTeX RIS

Andrey Sarantsev, Li-Cheng Tsai. 2016-08-01. Stationary Gap Distributions for Infinite Systems of Competing Brownian Particles. https://arxiv.org/abs/1608.00628

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