Consider a one-dimensional semi-infinite system of Brownian particles,
starting at Poisson(L) point process on the positive half-line, with
the left-most (Atlas) particle endowed a unit drift to the right.
We show that for the equilibrium density (L=2), the asymptotic
Gaussian space-time particle fluctuations are governed by the
stochastic heat equation with Neumann boundary condition at zero.
As a by product we resolve a conjecture of Pal and Pitman (2008)
about the asympotic (random) fBM trajectory of the Atlas particle.
In a complementary work, we derive and explicitly solve the Stefan
(free-boundary) equations for the limiting particle-profile when
starting at out of equilibrium density (L other than 2). We thus
determine the corresponding (non-random) asymptotic trajectory of
the Atlas particle.
This talk is based on joint works with Li-Cheng Tsai,
Manuel Cabezas, Andrey Sarantsev and Vladas Sidoravicius.