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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 thereflecting boundary at $x=0$. The interface is initially flat, $h(x,t=0)=0$. Wefocus 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) bodyof the distribution and the strongly asymmetric non-Gaussian tails. We findthat 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 tailexhibits 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 acompetition between two different fluctuation paths of the system. The fasterdecaying tail scales as$-\sqrt{t}\,\ln\mathcal{P}\simeq|H|^{5/2}f_{+}\left(L/\sqrt{|H|t}\right)$. Weevaluate the function $f_{+}(\dots)$ using a specially developed numericalmethod, which involves solving a nonlinear second-order elliptic equation inLagrangian coordinates. The faster-decaying tail also involves a sharptransition, 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 aring, and we believe that it has the same fractional order $5/2$. It issmoothed, however, by small diffusion effects.

Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line: the height statistics at a shifted point