On the chaotic character of the stochastic heat equation, II

TL;DR

This paper investigates the large-distance behavior of solutions to the stochastic heat equation with spatially-correlated Gaussian noise, revealing specific fluctuation exponents that depend on the noise's correlation decay rate.

Contribution

It establishes the fluctuation exponents for the solution in the Riesz-type correlation case, confirming physical predictions and extending Dalang's theory.

Findings

Fluctuation exponents are or space and 2or time, with rom 2/(4-)

Results hold for (x)\u2264 (x) with (x)(x) proportional to \u00f7x^{-\u007}

Findings align with earlier physical predictions.

Abstract

Consider the stochastic heat equation $\partial_t u = (\frac{\varkappa}{2})\Delta u+\sigma(u)\dot{F}$, where the solution $u:=u_t(x)$ is indexed by $(t,x)\in (0, \infty)\times\R^d$, 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 implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions.