  Consider a system of infinitely many Brownian particles on the real line. At
any moment, these particles can be ranked from the bottom upward. Each particle
moves as a Brownian motion with drift and diffusion coefficients depending on
its current rank. The gaps between consecutive particles form the
(infinite-dimensional) gap process. We find a stationary distribution for the
gap process. We also show that if the initial value of the gap process is
stochastically larger than this stationary distribution, this process converges
back to this distribution as time goes to infinity. This continues the work by
Pal and Pitman (2008). Also, this includes infinite systems with asymmetric
collisions, similar to the finite ones from Karatzas, Pal and Shkolnikov
(2016).
