Fractional moments of the Stochastic Heat Equation

Consider the solution $\mathcal{Z}(t,x)$ of the one-dimensional stochastic heat equation, with a multiplicative spacetime white noise, and with the delta initial data $\mathcal{Z}(0,x)=\delta (x)$. For any real $p>0$, we obtained detailed estimates of the $p$-th moment of $e^{t\mathrm{/}12}\mathcal{Z}(2t,0)$, as $t\to \mathrm{\infty }$, and from these estimates establish the one-point upper-tail large deviation principle of the Kardar-Parisi-Zhang equation. The deviations have speed $t$ and rate function ${\mathrm{\Phi }}_{+}(y)=\frac{4}{3}{y}^{3\mathrm{/}2}$. Our result confirms the existing physics predictions [Le Doussal, Majumdar, Schehr 16] and also [Kamenev, Meerson, Sasorov 16].