Journal of Theoretical Probability

, Volume 32, Issue 2, pp 545–585 | Cite as

Comparison Techniques for Competing Brownian Particles

  • Andrey SarantsevEmail author


Consider a finite system of Brownian particles on the real line. Each particle has drift and diffusion coefficients depending on its current rank relative to other particles, as in Karatzas et al. (Ann I H Poincare-PR 52(1):323–354, 2016). We prove some comparison results for these systems. As an example, we show that if we remove a few particles from the top, then the gaps between adjacent particles become stochastically larger, the local times of collision between adjacent particles become stochastically smaller, and the remaining particles shift upward, in the sense of stochastic ordering.


Reflected Brownian motion Competing Brownian particles Asymmetric collisions Skorohod problem Stochastic comparison 

Mathematics Subject Classification (2010)

Primary 60K35 Secondary 60J60 60J65 60H10 91B26 



I would like to thank Ioannis Karatzas, Soumik Pal, Xinwei Feng, Amir Dembo, and Vladas Sidoravicius for help and useful discussion. This research was partially supported by NSF grants DMS 1007563, DMS 1308340, DMS 1405210, and DMS 1409434.


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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of NevadaRenoUSA

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