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Consider an infinite collection of particles on the real line movingaccording to independent Brownian motions and such that the $i$-th particlefrom the left gets the drift $g_{i-1}$. The case where $g_0=1$ and $g_{i}=0$for all $i \in \mathbb{N}$ corresponds to the well studied infinite Atlasmodel. Under conditions on the drift vector $\mathbf{g} = (g_0, g_1, \ldots)'$it is known that the Markov process corresponding to the gap sequence of theassociated ranked particles has a continuum of product form stationarydistributions $\{\pi_a^{\mathbf{g}}, a \in S^{\mathbf{g}}\}$ where$S^{\mathbf{g}}$ is a semi-infinite interval of the real line. In this work weshow that all of these stationary distributions are extremal and ergodic. Wealso prove that any product form stationary distribution of this Markov processthat satisfies a mild integrability condition must be $\pi_a^{\mathbf{g}}$ forsome $a \in S^{\mathbf{g}}$. These results are new even for the infinite Atlasmodel. The work makes progress on the open problem of characterizing all theextremal invariant distributions of general competing Brownian particle systemsinteracting through their relative ranks. Proofs rely on synchronous and mirrorcoupling of Brownian particles and properties of the intersection local timesof the various particles in the infinite system.

Extremal Invariant Distributions of Infinite Brownian Particle Systems with Rank Dependent Drifts