Points of slow growth for parabolic SPDEs
Davar Khoshnevisan, Cheuk Yin Lee
TL;DR
This work analyzes points of slow growth for the parabolic SPDE ∂t u = ∂xx u + σ(u) ˙W with u(0)=1, establishing that random slow points exist despite a Lebesgue-null presence and that their distribution is governed by a universal boundary-crossing exponent λ(θ) independent of σ. By reducing the small-time behavior to a Gaussian linearization H and a controlled nonlinear remainder, the authors derive precise asymptotics and a multifractal structure for the slow-point set via localization techniques (H_α) and fractal dimensions (Hausdorff and Minkowski). They prove a sharp phase transition: the probability of finding slow points on a compact set K depends on λ(θ) relative to dim_H K and dim_M K, and they show that the slow-point set S(θ) has a dimension given by 1−2λ(θ) for θ≥θ_c, with a conjectured empty S(θ_c). Overall, the paper advances understanding of fine-scale temporal behavior in infinite-dimensional SPDEs and connects stochastic analysis with fractal geometry concepts.
Abstract
Consider the stochastic PDE, $\partial_tu = \partial^2_x u + σ(u) \dot{W}$ on $\mathbb{R}_+\times\mathbb{R}$, subject to $u(0)\equiv1$, where $\dot{W}$ denotes space-time white noise on $\mathbb{R}_+\times\mathbb{R}$ and $σ:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous. It is known that $u(t\,,x)-1$ has approximately a Gaussian distribution for every $x$ when $t\approx0$. Here we prove that there exist random points $x\in\mathbb{R}$ where the fluctuations of the solution near times zero are almost surely of sharp order $t^{1/4}$. Our work bears some loose resemblance to the study of the slow points of Brownian motion increments, though significant challenges arise due to the infinite-dimensional nature of the present problem.