Consider a stochastic interface h(x,t), described by the 1+1 Kardar-Parisi-Zhang (KPZ) equation on the half line x≥0. The interface is initially flat, h(x,t=0)=0, and driven by a Neumann boundary condition ∂xh(x=0,t)=A and by the noise. We study the short-time probability distribution PH,A,t of the one-point height H=h(x=0,t). Using the optimal fluctuation method, we show that -lnPH,A,t scales as t-1/2sH,At1/2. For small and moderate |A| this more general scaling reduces to the familiar simple scaling -lnPH,A,t≃t-1/2s(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.
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We are grateful to Naftali Smith for a valuable advice and a critical reading of the manuscript. B.M. acknowledges financial support from the Israel Science Foundation (Grant No. 807/16).
© 2018 American Physical Society.