Fluctuations of stochastic PDEs with long-range correlations

We study the large-scale dynamics of the solution to a nonlinear stochastic heat equation (SHE) in dimensions d≥3 with long-range dependence. This equation is driven by multiplicative Gaussian noise, which is white in time and coloured in space with non-integrable spatial covariance that decays at the rate of |x|−κ at infinity, where κ∈(2,d). Inspired by recent studies on SHE and KPZ equations driven by noise with compactly supported spatial correlation, we demonstrate that the correlations persist in the large-scale limit. The fluctuations of the diffusively scaled solution converge to the solution of a stochastic heat equation with additive noise whose correlation is the Riesz kernel of degree −κ. Moreover, the fluctuations converge as a distribution-valued process in the optimal Hölder topologies.