On the space-time fluctuations of the SHE and KPZ equation in the entire L²-regime for spatial dimensions d ≥ 3

2024/02/10 by Te-Chun Wang, Wang, Te-Chun
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Advanced Mathematical Physics Problems #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2402.06874

Abstract

We consider the mollified versions of the Kardar-Parisi-Zhang (KPZ) equation and the stochastic heat equation (SHE) in high dimensions d≥ 3 and analyze their probability distributions as the mollification is removed. Up to the L²-criticality, we prove Gaussian limits, possibly with random perturbations, for the space-time fluctuations of the mollified versions around their stationarity. This result establishes a continuous analogue of the discrete case obtained by Cosco and Nakajima (2021), and further extends it to a multi-point framework.

Related

• Space-time fluctuation of the Kardar-Parisi-Zhang equation in d≥ 3 and the Gaussian free field
• Equivalence of fluctuations of discretized SHE and KPZ equations in the subcritical weak disorder regime
• Law of large numbers and fluctuations in the sub-critical and L²\n regions for SHE and KPZ equation in dimension d\≥ 3
• Fluctuations of stochastic PDEs with long-range correlations
• Scaling limit of the KPZ equation with non-integrable spatial correlations