  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 does not have a unique stationary
distribution and in fact for every $a \ge 0$, $\pi_a :=
\bigotimes_{i=1}^{\infty} \operatorname{Exp}(2 + ia)$ is a stationary measure
for the gap process. We say that an initial distribution of gaps is in the weak
domain of attraction of the stationary measure $\pi_a$ if the time averaged
laws of the stochastic process of the gaps, when initialized using that
distribution, converge to $\pi_a$ 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 $\pi_a$ for each
$a\ge 0$. The cases $a=0$ and $a>0$ are qualitatively different as is seen from
the analysis and the sufficient conditions that we provide. Proofs are based on
the analysis of synchronous couplings, namely, couplings of the ranked particle
systems started from different initial configurations, but driven using the
same set of Brownian motions.
