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Brownian Particles with Rank‐Dependent Drifts: Out‐of‐Equilibrium Behavior
Author(s) -
Cabezas M.,
Dembo A.,
Sarantsev A.,
Sidoravicius V.
Publication year - 2019
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21825
Subject(s) - brownian motion , mathematics , countable set , range (aeronautics) , convergence (economics) , particle (ecology) , rank (graph theory) , interacting particle system , particle system , boundary (topology) , constant (computer programming) , statistical physics , weak convergence , mathematical analysis , stochastic differential equation , physics , statistics , combinatorics , materials science , oceanography , computer security , continuous time stochastic process , geology , computer science , economics , composite material , asset (computer security) , programming language , economic growth , operating system
Abstract We study the long‐range asymptotic behavior for an out‐of‐equilibrium, countable, one‐dimensional system of Brownian particles interacting through their rank‐dependent drifts. Focusing on the semi‐infinite case, where only the leftmost particle gets a constant drift to the right, we derive and solve the corresponding one‐sided Stefan (free‐boundary) equations. Via this solution we explicitly determine the limiting particle‐density profile as well as the asymptotic trajectory of the leftmost particle. While doing so we further establish stochastic domination and convergence to equilibrium results for the vector of relative spacings among the leading particles. © 2019 Wiley Periodicals, Inc.