Chung's LIL for the linear stochastic fractional heat equation at origin

TL;DR

The paper analyzes Chung's law of the iterated logarithm at the initial time for the linear stochastic fractional heat equation $\partial_t u=-(-\Delta)^{\alpha/2}u+\dot W$ with $\alpha\in(1,2]$ and a Gaussian noise $\dot W$ white in time and spatially a fractional Brownian motion with $H\in((2-\alpha)/2,1)$. It leverages a decomposition of the solution into Gaussian components and a localization approach to handle the boundary time $t=0$, establishing that for fixed $x$, $\liminf_{\varepsilon\downarrow0}\sup_{t\in[0,\varepsilon]} |u(t,x)| / (\varepsilon^{\theta}(\log\log \varepsilon^{-1})^{- heta}) = \kappa\lambda^{\theta}$ a.s., where $\theta=\tfrac12-(1-H)/\alpha$ and $\lambda$ is the small-ball constant of a fractional Brownian motion with index $\theta$. The lower bound follows from small-ball probabilities, while the upper bound uses a localization argument with a family of localized Gaussian fields $u_n$, together with sharp small-ball estimates for these fields. The results extend Chung LIL analysis to the initial time for the fractional SPDE regime and clarify the fine short-time regularity of the SFHE solution.

Abstract

Consider the linear stochastic fractional heat equation with vanishing initial condition: $$ \frac{\partial u (t,x)}{\partial t}=-(-Δ)^{\fracα2}u (t,x) + \dot{W}(t,x),\quad t> 0,\, x\in \mathbb R, $$ where $-(-Δ)^{\fracα{2}}$ denotes the fractional Laplacian with power $α\in (1,2]$, and the driving noise $\dot W$ is a centered Gaussian field which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in\left(\frac {2-α}2,1\right)$. We establish Chung's law of the iterated logarithm for the solution at $t=0$.

Theorems & Definitions (16)

• Theorem 1.1
• Proposition 2.1
• Proposition 2.1
• proof
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof : Proof of Theorem \ref{['pr:Chung:u']}
• Lemma 3.1
• proof