  We consider the weakly asymmetric limit of simple exclusion process with
drift to the left, starting from step Bernoulli initial data with
$\rho_-<\rho_+$ so that macroscopically one has a rarefaction fan. We study the
fluctuations of the process observed along slopes in the fan, which are given
by the Hopf--Cole solution of the Kardar-Parisi-Zhang (KPZ) equation, with
appropriate initial data. For slopes strictly inside the fan, the initial data
is a Dirac delta function and the one point distribution functions have been
computed in [Comm. Pure Appl. Math. 64 (2011) 466-537] and [Nuclear Phys. B 834
(2010) 523-542]. At the edge of the rarefaction fan, the initial data is
one-sided Brownian. We obtain a new family of crossover distributions giving
the exact one-point distributions of this process, which converge, as
$T\nearrow\infty$ to those of the Airy $\mathcal{A}_{2\to \mathrm{BM}}$
process. As an application, we prove moment and large deviation estimates for
the equilibrium Hopf-Cole solution of KPZ. These bounds rely on the apparently
new observation that the FKG inequality holds for the stochastic heat equation.
Finally, via a Feynman-Kac path integral, the KPZ equation also governs the
free energy of the continuum directed polymer, and thus our formula may also be
interpreted in those terms.
