Lyapunov exponents of the half-line SHE

Abstract

We consider the half-line stochastic heat equation (SHE) with Robin boundary parameter $A=-\frac{1}{2}$. Under narrow wedge initial condition, we compute every positive (including non-integer) Lyapunov exponents of the half-line SHE. As a consequence, we prove a large deviation principle for the upper tail of the half-line KPZ equation under Neumann boundary parameter $A=-\frac{1}{2}$ with rate function $\hat{I}{¦}_{+}^{\text{hf}}(s)=\frac{2}{3}{s}^{\frac{3}{2}}$. This confirms the prediction of [Krajenbrink and Le Doussal 2018] and [Meerson, Vilenkin 2018] for the upper tail exponent of the half-line KPZ equation.

25 pages