On the multifractal local behavior of parabolic stochastic PDEs

Abstract

Consider the stochastic heat equation $\dot{u}=\frac12 u"+\sigma(u)\xi$ on $(0\,,\infty)\times\mathbb{R}$ subject to $u(0)\equiv1$, where $\sigma:\mathbb{R}\to\mathbb{R}$ is a Lipschitz (local) function that does not vanish at $1$, and $\xi$ denotes space-time white noise. It is well known that $u$ has continuous sample functions; as a result, $\lim_{t\downarrow0}u(t\,,x)= 1$ almost surely for every $x\in\mathbb{R}$. The corresponding fluctuations are also known: For every fixed $x\in\mathbb{R}$, $t\mapsto u(t\,,x)$ looks locally like a fixed multiple of fractional Brownian motion (fBm) with index $1/4$. 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 $t\mapsto u(t\,,x)$ 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 $t\mapsto u(t\,,x)$ 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 Khoshnevisan, Kim and Xiao (2016). To the best of our knowledge, the short-time results of the present paper are observed here for the first time.