  We consider a stochastic interface $h(x,t)$, described by the $1+1$
Kardar-Parisi-Zhang (KPZ) equation on the half-line $x\geq0$ with the
reflecting boundary at $x=0$. The interface is initially flat, $h(x,t=0)=0$. We
focus on the short-time probability distribution
$\mathcal{P}\left(H,L,t\right)$ of the height $H$ of the interface at point
$x=L$. Using the optimal fluctuation method, we determine the (Gaussian) body
of the distribution and the strongly asymmetric non-Gaussian tails. We find
that the slower-decaying tail scales as
$-\sqrt{t}\,\ln\mathcal{P}\simeq\left|H\right|^{3/2}f_{-}\left(L/\sqrt{\left|H\right|t}\right)$,
and calculate the function $f_{-}(\dots)$ analytically. Remarkably, this tail
exhibits a first-order dynamical phase transition at a critical value of $L$,
$L_{c}=0.60223\dots\sqrt{\left|H\right|t}$. The transition results from a
competition between two different fluctuation paths of the system. The faster
decaying tail scales as
$-\sqrt{t}\,\ln\mathcal{P}\simeq|H|^{5/2}f_{+}\left(L/\sqrt{|H|t}\right)$. We
evaluate the function $f_{+}(\dots)$ using a specially developed numerical
method, which involves solving a nonlinear second-order elliptic equation in
Lagrangian coordinates. The faster-decaying tail also involves a sharp
transition, which occurs at a critical value $L_{c}\simeq2\sqrt{2|H|t}/\pi$.
This transition is similar to the one recently found for the KPZ equation on a
ring, and we believe that it has the same fractional order $5/2$. It is
smoothed, however, by small diffusion effects.
