On the chaotic character of the stochastic heat equation, II

英文摘要：

Consider the stochastic heat equation $\partial_t u = (\frac{\varkappa}{2})Δu+σ(u)\dot{F}$, 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 $f(x)\propto \|x\|^{-α}$, then the "fluctuation exponents" of the solution are $ψ$ for the spatial variable and $2ψ-1$ for the time variable, where $ψ:=2/(4-α)$. Moreover, these exponent relations hold as long as $α\in(0, d\wedge 2)$; that is precisely when Dalang's theory implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions.