A Macroscopic Multifractal Analysis of Parabolic Stochastic PDEs

Abstract

It is generally argued that the solution to a stochastic PDE with multiplicative noise-such asu = 1 2 u ′′ + uξ, where ξ denotes space-time white noise-routinely produces exceptionally-large peaks that are "macroscopically multifractal." See, for example, Gibbon and Doering (2005), Gibbon and Titi (2005), and Zimmermann et al (2000). A few years ago, we proved that the spatial peaks of the solution to the mentioned stochastic PDE indeed form a random multifractal in the macroscopic sense of Barlow and Taylor (1989;. The main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio-temporal peaks of the solution form infinitely-many different multifractals on infinitely-many different scales, which we sometimes refer to as "stretch factors." A simpler, though still complex, such structure is shown to also exist for the constant-coefficient version of the said stochastic PDE.

On the multifractal local behavior of parabolic stochastic PDEs

“…In particular, an application of Fubini's theorem implies that, on an x-set of full Lebesgue measure, the short-time behavior of the peaks of the random function t → u(t , x) are governed by the law of the iterated logarithm for fBm, up to possibly a suitable normalization constant. By contrast, the main result of this paper claims that, on an x-set of full Hausdorff dimension, the short-time peaks of t → u(t , x) follow a non-iterated logarithm law, and that those peaks contain a rich multifractal structure a.s.Large-time variations of these results were predicted in the physics literature a number of years ago and proved very recently in [10,11]. To the best of our knowledge, the short-time results of the present paper are observed here for the first time.…”

“…It has recently been shown by Kunwoo Kim, Yimin Xiao, and the second author [10,11] that the largest global oscillations of the random map (t , x) → u(t , x) form an asymptotic multifractal; this property had been predicted earlier in a voluminous physics literature on the present "intermittent" stochastic systems [as well as for more complex systems]. It is not hard to deduce from Theorem 1.1 that the high local oscillations of (t , x) → u(t , x) are also multifractal in a precise sense that we now explain next.…”

Fractal Geometry of the Valleys of the Parabolic Anderson Equation

“…[CM94]) of the parabolic Anderson equation. Motivated by this result and its analogue in other stochastic PDEs with multiplicative noise, [KKX17,KKX18] studied the macroscopic fractality of the spatial-temporal tall peaks of the solution for a large collection of parabolic stochastic PDEs including the parabolic Anderson equation. They had shown that the values of the macroscopic Hausdorff dimension of the tall peaks are distinct and nontrivial when the length scale and stretch factor vary, a property which symbolizes the multifractality.…”

“…We seek to study the fractal nature of the level sets S β (V(γ)) as t, x get large. The fractal nature of the peaks in parabolic Anderson equation had been quantified in [KKX17,KKX18] by the Barlow-Taylor's macroscopic Hausdorff dimension. Motivated by those works, we aim to determine the macroscopic Hausdorff dimension of S β (V(γ)).…”

The almost-sure asymptotic behavior of the solution to the stochastic heat equation with Lévy noise

“…However, the translation of this purely moment-based notion of intermittency to a pathwise description of the exponentially large peaks of the solution, sometimes referred to as physical intermittency, has not been fully resolved yet; see [2,Section 2.4] or [16,Chapter 7.1] for some heuristic arguments. Despite recent results of [18] on the multifractal nature of the spacetime peaks of the solution, the exact almost-sure asymptotics of the solution as time tends to infinity, for fixed spatial location, are still unknown. To our best knowledge, only a weak law of large numbers has been proved rigorously for certain initial conditions when σ is a linear function; see [3] and, in particular, [1,10], where much deeper fluctuation results were obtained.…”