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We study interacting systems of linear Brownian motions whose drift vector atevery time point is determined by the relative ranks of the coordinateprocesses at that time. Our main objective has been to study the long rangebehavior of the spacings between the Brownian motions arranged in increasingorder. For finitely many Brownian motions interacting in this manner, wecharacterize drifts for which the family of laws of the vector of spacings istight, and show its convergence to a unique stationary joint distribution givenby independent exponential distributions with varying means. We also study oneparticular countably infinite system, where only the minimum Brownian particlegets a constant upward drift, and prove that independent and identicallydistributed exponential spacings remain stationary under the dynamics of such aprocess. Some related conjectures in this direction have also been discussed.

One-dimensional Brownian particle systems with rank dependent drifts