On the chaotic character of the stochastic heat equation, II

Details On the chaotic character of the stochastic heat equation, II On the chaotic character of the stochastic heat equation, II

2011-11-21

arxiv.org/abs/1111.4728v1

Mathematics

Probability

Scientific paper

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.

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