Fractional moments of the stochastic heat equation

Consider the solution Z(t,x) of one-dimensional stochastic heat equation, with a multiplicative spacetime white noise, and delta initial data Z(0,x)=?(x). For any real p>0, we obtained detailed estimates pth moment et/12Z(2t,0), as t??, from these establish one-point upper-tail large deviation principle Kardar–Parisi–Zhang equation. The deviations have speed t rate function ?+(y)=43y3/2. Our result confirms existing physics predictions [Europhys. Lett. 113 (2016) 60004] also [Phys. Rev. E 94 032108].