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.
From: Cheuk Yin Lee [view email]
[v1]
Wed, 17 Dec 2025 08:19:40 UTC (28 KB)
[v2]
Mon, 10 Aug 2026 04:46:10 UTC (28 KB)
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