On the chaotic character of the stochastic heat equation, II

Abstract

Consider the stochastic heat equation ${\partial_t u = (\varkappa/2)\Delta u+\sigma(u)\dot{F}}$, where the solution u := u _t (x) is indexed by ${(t, x) \in (0, \infty) \times {\bf R}^d}$, and ${\dot{F}}$ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large-${\|x\|}$ fixed-t behavior of the solution u in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function f of the noise is of Riesz type, that is ${f(x)\propto \|x\|^{-\alpha}}$, then the “fluctuation exponents” of the solution are ${\psi}$ for the spatial variable and ${2\psi-1}$ for the time variable, where ${\psi:=2/(4-\alpha)}$. Moreover, these exponent relations hold as long as ${\alpha \in (0, d \wedge 2)}$ ; that is precisely when Dalang’s theory [Dalang, Electron J Probab 4:(Paper no. 6):29, 1999] implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions [Kardar et al., Phys Rev Lett 58(20):889–892, 1985; Kardar and Zhang, Phys Rev Lett 58(20):2087–2090, 1987].