Consider the stochastic heat equation ∂tu=(2ϰ)Δu+σ(u)F˙, where the solution u:=ut(x)
is indexed by (t,x)∈(0,∞)×Rd, and 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)∝∥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∈(0,d∧2); that is precisely
when Dalang's theory implies the existence of a solution to our stochastic PDE.
These findings bolster earlier physical predictions