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We consider a finite or countable collection of one-dimensional Brownianparticles whose dynamics at any point in time is determined by their rank inthe entire particle system. Using Transportation Cost Inequalities forstochastic processes we provide uniform fluctuation bounds for the orderedparticles, their local time of collisions, and various associated statisticsover intervals of time. For example, such processes, when exponentiated andrescaled, exhibit power law decay under stationarity; we derive concentrationbounds for the empirical estimates of the index of the power law over largeintervals of time. A key ingredient in our proofs is a novel upper bound on theLipschitz constant of the Skorokhod map that transforms a multidimensionalBrownian path to a path which is constrained not to leave the positive orthant.

Concentration of measure for systems of Brownian particles interacting through their ranks