Uniform dimension theorems for parabolic SPDEs

TL;DR

This paper establishes uniform Hausdorff dimension results for solutions to certain stochastic PDEs on a torus, revealing how the solution's fractal properties relate to the initial data and noise structure.

Contribution

It proves dimension theorems for parabolic SPDEs, extending results to lower dimensions under specific conditions on the noise coefficient matrix.

Findings

Dimension equality holds for all compact sets in space for p≥4.

If noise coefficient is constant and invertible, the result holds for all p≥2.

The Hausdorff dimension of the solution's image doubles that of the initial set.

Abstract

Consider the following $p$-dimensional system of It\^o type stochastic PDEs, \begin{align*}\left[\begin{aligned} &\partial_t u(t\,,x) = \partial^2_x u(t\,,x) + b(u(t\,,x)) + \sigma(u(t\,,x)) \xi(t\,,x)\\ &\text{for $(t\,,x)\in(0\,,\infty)\times\mathbb{T}$, subject to $u(0) \equiv u_0$ on $\mathbb{T}$}, \end{aligned}\right.\end{align*} where $\mathbb{T}$ denotes a given one-dimensional torus, the initial data $u_0:\mathbb{T}\to\mathbb{R}^p$ is assumed to be fixed and non-random and in $C^{1/2}(\mathbb{T}\,;\mathbb{R}^p)$, and $\xi$ denotes a $p$-dimensional space-time white noise. Under certain regularity conditions on $b$ and $\sigma$, it is proved that, if $p \ge 4$, then \begin{equation*} \mathrm{P}\{\operatorname{dim_{_H}} u(\{t\}\times F) = 2\operatorname{dim_{_H}} F \ \text{$\forall$compact $F\subset\mathbb{T}$, $t>0$}\}=1. \end{equation*} If in addition the matrix $\sigma(v)$ does not depend on $v\in\mathbb{R}^p$, and is nonsingular, then the above equality holds for all $p\ge2$.