## One-dimensional Brownian particle systems with rank-dependent drifts.(English)Zbl 1166.60061

The paper studies interacting systems of linear Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate process at that time. Loose speaking, the Brownian particles evolve independently, except that the $$i$$’th-ranked particle is given drift $$\delta_i$$. The main objective of the paper is to characterize the long-range behaviour of the spacings between the Brownian motions arranged in increasing order. For finite particle systems necessary and sufficient conditions on the drifts $$\delta_1,\dotsc,\delta_N$$ for the spacing between the highest and the lowest particle to be tight over $$t \geq 0$$ is given. In the affirmative case, the distribution of the spacings is shown to converge in total variation norm to a unique stationary distribution given by independent exponential variables with specified rates. In particular, this holds for the so-called Atlas model, which is given by $$\delta_1 = \delta > 0$$ and $$\delta_2=\dotsc=\delta_N=0$$. The second part of the paper studies the infinite dimensional Atlas model, i.e. countable infinite many Brownian motions interacting through ranks via $$\delta_1 = \delta > 0$$ and $$0=\delta_2=\dotsc$$. It is proven, that independent and identically distributed exponential spacings remain stationary under the dynamics of such a model. Some related conjectures in this direction are also discussed. The paper is thorough and detailed, e.g. discussing existence and uniqueness of such particle systems, and an extensive review of the literature is also given.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60F05 Central limit and other weak theorems 60G15 Gaussian processes 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J65 Brownian motion
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