  We study the solution of the Kardar-Parisi-Zhang (KPZ) equation for the
stochastic growth of an interface of height $h(x,t)$ on the positive half line,
equivalently the free energy of the continuum directed polymer in a half space
with a wall at $x=0$. The boundary condition $\partial_x h(x,t)|_{x=0}=A$
corresponds to an attractive wall for $A<0$, and leads to the binding of the
polymer to the wall below the critical value $A=-1/2$. Here we choose the
initial condition $h(x,0)$ to be a Brownian motion in $x>0$ with drift
$-(B+1/2)$. When $A+B \to -1$, the solution is stationary, i.e. $h(\cdot,t)$
remains at all times a Brownian motion with the same drift, up to a global
height shift $h(0,t)$. We show that the distribution of this height shift is
invariant under the exchange of parameters $A$ and $B$. For any $A,B > - 1/2$,
we provide an exact formula characterizing the distribution of $h(0,t)$ at any
time $t$, using two methods: the replica Bethe ansatz and a discretization
called the log-gamma polymer, for which moment formulae were obtained. We
analyze its large time asymptotics for various ranges of parameters $A,B$. In
particular, when $(A, B) \to (-1/2, -1/2)$, the critical stationary case, the
fluctuations of the interface are governed by a universal distribution akin to
the Baik-Rains distribution arising in stationary growth on the full-line. It
can be expressed in terms of a simple Fredholm determinant, or equivalently in
terms of the Painlev\'e II transcendent. This provides an analog for the KPZ
equation, of some of the results recently obtained by Betea-Ferrari-Occelli in
the context of stationary half-space last-passage-percolation. From
universality, we expect that limiting distributions found in both models can be
shown to coincide.
