Uniform dimension theorems for parabolic SPDEs

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)\ &$\text{for }(t,x)\in (0,\mathrm{\infty })\times \mathbb{T}\text{, subject to }u(0)\equiv u_0\text{ on }\mathbb{T}$, \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 $C^{1\mathrm{/}2}(\mathbb{T};{\mathbb{R}}^p)$, 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 \ $\mathrm{\forall }\text{compact }F\subset \mathbb{T}\text{, }t>0$}=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 $p\ge 2$.