Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line

Consider a stochastic interface $h(x,t)$, described by the $1+1$ Kardar-Parisi-Zhang (KPZ) equation on the half-line $x\geq 0$. The interface is initially flat, $h(x,t=0)=0$, and driven by a Neumann boundary condition $\partial_x h(x=0,t)=A$ and by the noise. We study the short-time probability distribution $\mathcal{P}\left(H,A,t\right)$ of the one-point height $H=h(x=0,t)$. Using the optimal fluctuation method, we show that $-\ln \mathcal{P}\left(H,A,t\right)$ scales as $t^{-1/2} s \left(H,A t^{1/2}\right)$. For small and moderate $|A|$ this more general scaling reduces to the familiar simple scaling $-\ln \mathcal{P}\left(H,A,t\right)\simeq t^{-1/2} s(H)$, where $s$ is independent of $A$ and time and equal to one half of the corresponding large-deviation function for the full-line problem. For large $|A|$ we uncover two asymptotic regimes. At very short time the simple scaling is restored, whereas at intermediate times the scaling remains more general and $A$-dependent. The distribution tails, however, always exhibit the simple scaling in the leading order.