Electronic Journal of Probability
- Electron. J. Probab.
- Volume 22 (2017), paper no. 56, 20 pp.
Stationary gap distributions for infinite systems of competing Brownian particles
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) [PP08] in the negative. This result is further generalized for infinite systems of competing Brownian particles with generic rank-based drifts.
Electron. J. Probab., Volume 22 (2017), paper no. 56, 20 pp.
Received: 1 February 2017
Accepted: 18 June 2017
First available in Project Euclid: 5 July 2017
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Sarantsev, Andrey; Tsai, Li-Cheng. Stationary gap distributions for infinite systems of competing Brownian particles. Electron. J. Probab. 22 (2017), paper no. 56, 20 pp. doi:10.1214/17-EJP78. https://projecteuclid.org/euclid.ejp/1499220068