  We consider the one-dimensional Kardar-Parisi-Zhang (KPZ) equation with half
Brownian motion initial condition, studied previously through the weakly
asymmetric simple exclusion process. We employ the replica Bethe ansatz and
show that the generating function of the exponential moments of the height is
expressed as a Fredholm determinant. From this the height distribution and its
asymptotics are studied. Furthermore using the replica method we also discuss
the multi-point height distribution. We find that some nice properties of the
deformed Airy functions play an important role in the analysis.
