Points of slow growth for parabolic SPDEs

TL;DR

This paper investigates the initial fluctuation behavior of solutions to a stochastic heat equation with white noise, identifying points where fluctuations grow at a specific slow rate of t^{1/4}.

Contribution

It establishes the existence of random spatial points where the solution's fluctuations near time zero are of order t^{1/4}, revealing new slow growth phenomena.

Findings

Existence of points with fluctuations of order t^{1/4} near time zero.

Fluctuations are almost surely sharp at these points.

The work connects to slow points in Brownian motion increments.

Abstract

Consider the stochastic PDE, $\partial_tu = \partial^2_x u + \sigma(u) \dot{W}$ 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 $\sigma:\mathbb{R}\to\mathbb{R}$ 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.