Fractional moments of the Stochastic Heat Equation

TL;DR

This paper derives detailed asymptotic estimates for fractional moments of the stochastic heat equation's solution, confirming physics-based predictions and establishing a large deviation principle for the KPZ equation's upper tail.

Contribution

It provides rigorous estimates of fractional moments and confirms the large deviation rate function for the KPZ equation's upper tail, aligning with prior physics predictions.

Findings

Established the large deviation principle with rate function (y)=rac{4}{3}y^{3/2}

Provided detailed asymptotic estimates for fractional moments of the solution

Confirmed physics predictions regarding the upper tail behavior

Abstract

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 $e^{t/12}\mathcal{Z}(2t,0)$, as $t\to\infty$, 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 $\Phi_+(y)=\frac{4}{3}y^{3/2}$. Our result confirms the existing physics predictions [Le Doussal, Majumdar, Schehr 16] and also [Kamenev, Meerson, Sasorov 16].