2023/03/17 by Luca Gerolla, Martin Hairer, Gerolla, Luca +3 · 2 citations
Mathematics · Physics and Astronomy · Economics, Econometrics and Finance · #Stochastic processes and statistical mechanics #Advanced Thermodynamics and Statistical Mechanics #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2303.09811
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.