On the chaotic character of the stochastic heat equation, II

Consider the stochastic heat equation ∂^t u = (x/2)Δ u+σ(u)Ḟ, where the solution u:= u _t(x) is indexed by (t, x) ∈ (0, ∞) × ℝ^d, and Ḟ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large-{double pipe}x{double pipe} 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)∝ {double pipe}x{double pipe}^-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 {n-ary logical and} 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].