Uniform dimension theorems for parabolic SPDEs
Davar Khoshnevisan, Cheuk Yin Lee, Fei Pu, Yimin Xiao
TL;DR
This work analyzes a $p$-component parabolic SPDE on the circle driven by space-time white noise and proves uniform Hausdorff-dimension formulas for spatial traces of the solution. It develops a Gaussian-linearization framework around small times, establishes strong local nondeterminism for the additive heat field, and then extends to nonlinear and multiplicative noise via truncation and Girsanov, aided by a polarity condition. A general criterion based on Hölder regularity and small-ball counts yields uniform dimension results that apply across time, space, and space-time, including a torus version under degeneracy-control assumptions. The results connect to Kaufman’s uniform-dimension theorem for Brownian motion and reveal fractal properties of SPDE solutions under high-dimensional noise, with sharp dimension thresholds depending on the dimension $p$ and the diffusion structure $\sigma$. These findings advance the understanding of the geometric structure of SPDE solutions and their dependence on noise and nonlinearity.
Abstract
Consider the following $p$-dimensional system of Itô type stochastic PDEs, \begin{align*}\left[\begin{aligned} &\partial_t u(t\,,x) = \partial^2_x u(t\,,x) + b(u(t\,,x)) + σ(u(t\,,x)) ξ(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 $ξ$ denotes a $p$-dimensional space-time white noise. Under certain regularity conditions on $b$ and $σ$, 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 $σ(v)$ does not depend on $v\in\mathbb{R}^p$, and is nonsingular, then the above equality holds for all $p\ge2$.