A LANDSCAPE OF PEAKS: THE INTERMITTENCY ISLANDS OF THE STOCHASTIC HEAT EQUATION WITH LeVY NOISE

• 成果类型: Article
• 署名作者: Chong, Carsten; Kevei, Peter
• 署名单位: Columbia University; Szeged University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1623
• 发表日期: 2023
• 页码: 1449-1501
• 关键词: wave-equation; driven; dimension; BEHAVIOR; moments

摘要

We show that the spatial profile of the solution to the stochastic heat equation features multiple layers of intermittency islands if the driving noise is non-Gaussian. On the one hand, as expected, if the noise is sufficiently heavy-tailed, the largest peaks of the solution will be taller under multiplica-tive than under additive noise. On the other hand, surprisingly, as soon as the noise has a finite moment of order d2, where dis the spatial dimension, the largest peaks will be of the same order for both additive and multiplicative noise, which is in sharp contrast to the behavior of the solution under Gaus-sian noise. However, in this case a closer inspection reveals a second layer of peaks, beneath the largest peaks, that is exclusive to multiplicative noise and that can be observed by sampling the solution on the lattice. Finally, we compute the macroscopic Hausdorff and Minkowski dimensions of the inter-mittency islands of the solution. Under both additive and multiplicative noise, if it is not too heavy-tailed, the largest peaks will be self-similar in terms of their large-scale multifractal behavior. But under multiplicative noise, this type of self-similarity is not present in the peaks observed on the lattice.