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 \(\Phi _+^{\text {hf}} (s) = \frac{2}{3} s^{\frac{3}{2}}\). This confirms the prediction of [44, 52] for the upper tail exponent of the half-line KPZ equation.

中文翻译：

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半线SHE的Lyapunov指数

我们考虑具有Robin边界参数\（A =-\ frac {1} {2} \）的半线随机热方程（SHE ）。在窄楔形初始条件下，我们计算半线SHE的每个正（包括非整数）Lyapunov指数。结果，我们证明了诺伊曼边界参数\（A =-\ frac {1} {2} \）下半线KPZ方程上尾的大偏差原理，其速率函数为\（\ Phi _ + ^ {\ text {hf}}（s）= \ frac {2} {3} s ^ {\ frac {3} {2}} \）。这证实了对半线KPZ方程的上尾指数的[44，52]预测。