Intermittency and multifractality: a case study via parabolic stochastic PDEs

DOI10.1214/16-AOP1147zbMATH Open1418.60081arXiv1503.06249OpenAlexW2133203656MaRDI QIDQ682245

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Publication date: 14 February 2018

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Abstract: Let $xi$ denote space-time white noise, and consider the following stochastic partial differential equations: (i) $$, started identically at one; and (ii) $$, started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in different universality classes. We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic PDEs on $mathbf{R}_{+}imesmathbf{R}^d$ with $dge2$. G. Lawler has asked us if intermittency is the same as multifractality. The present work gives a negative answer to this question. As a byproduct of our methods, we prove also that the peaks of the Brownian motion form a large-scale monofractal, whereas the peaks of the Ornstein--Uhlenbeck process on $mathbfR$ are multifractal. Throughout, we make extensive use of the macroscopic fractal theory of M.T. Barlow and S. J. Taylor (1989, 1992). We expand on aspects of the Barlow-Taylor theory, as well.

Full work available at URL: https://arxiv.org/abs/1503.06249

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