arXiv subjects

Luca Gerolla

Publications and source records attributed to Luca Gerolla.

2 recordsLinked to original sources

arXiv2024

Scaling limit of the KPZ equation with non-integrable spatial correlations

We study the large scale fluctuations of the KPZ equation in dimensions $d \geq 3$ driven by Gaussian noise that is white in time Gaussian but features non-integrable spatial correlation with decay rate $\kappa \in (2, d)$ and a suitable limiting profile. We show that its scaling limit is described by the corresponding additive stochastic heat equation. In contrast to the case of compactly supported covariance, the noise in the stochastic heat equation retains spatial correlation with covariance $|x|^{-\kappa}$. Surprisingly, the noise driving the limiting equation turns out to be the scaling limit of the noise driving the KPZ equation so that, under a suitable coupling, one has convergence in probability, unlike in the case of integrable correlations where fluctuations are enhanced in the limit and convergence is necessarily weak.

math.PR↗

arXiv2023

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 $d \geq 3$ 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.

math.PR↗