Abstract

The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in the infinite Atlas model has a one parameter family {p(a), a > 0} of product form mutually singular stationary distributions. We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure p(a) if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to p(a) weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of p(a) for each a. Results on extremality and ergodicity of p(a) will be presented. Finally, time permitting, we will describe some recent results on fluctuations of the Atlas model from inhomogeneous stationary profiles.  This is based on joint work with Sayan Banerjee and Peter Rudzis.