Fractional moments of the stochastic heat equation

Consider the solution Z(t, x) of the one-dimensional stochastic heat equation, with a multiplicative spacetime white noise, and with the delta initial data Z(0, x) = delta(x). For any real p > 0, we obtained detailed estimates of the pth moment of e(t/12)Z(2t, 0), as t -> infinity, 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) = 4/3 y(3/2). Our result confirms the existing physics predictions [Europhys. Lett. 113 (2016) 60004] and also [Phys. Rev. E 94 (2016) 032108].