Fluctuations of stochastic PDEs with long-range correlations

TL;DR

This paper investigates the large-scale behavior of solutions to a nonlinear stochastic heat equation with long-range spatial correlations, revealing persistent correlations and convergence to a limiting stochastic heat equation with Riesz kernel-based noise.

Contribution

It demonstrates that long-range spatial correlations in the noise persist in the large-scale limit, leading to a new form of the limiting stochastic heat equation with Riesz kernel noise.

Findings

Fluctuations converge to a stochastic heat equation with Riesz kernel noise.

Correlations in the noise persist at large scales.

Convergence occurs in optimal H"older topologies.

Abstract

We study the large-scale dynamics of the solution to a nonlinear stochastic heat equation (SHE) in dimensions $d \geq 3$ 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 $|x|^{-\kappa}$ at infinity, where $\kappa \in (2, d)$. 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 $-\kappa$. Moreover, the fluctuations converge as a distribution-valued process in the optimal H\"older topologies.