Growth rates for the H\"older coefficients of the linear stochastic fractional heat equation with rough dependence in space

TL;DR

This paper analyzes the growth rates of H"older coefficients for solutions to a linear stochastic fractional heat equation driven by rough spatial dependence, providing exact asymptotics and sharp bounds.

Contribution

It establishes precise asymptotic growth rates for the solution and H"older coefficients of a fractional stochastic heat equation with rough spatial noise.

Findings

Derived exact asymptotics for the solution as time and space tend to infinity.

Established sharp growth rates for H"older coefficients.

Applied Talagrand's and Sudakov's theorems to analyze the solution.

Abstract

We study the linear stochastic fractional heat equation $$ \frac{\partial}{\partial t}u(t,x)=-(-\Delta)^{\frac{\alpha}2}u (t,x)+\dot{W}(t,x),\ \ t> 0,\ \ x\in\RR, $$ where $-(-\Delta)^{\frac{\alpha}{2}}$ denotes the fractional Laplacian with power $\alpha\in (1, 2)$, 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 $H\in\left(\frac {2-\alpha}2,\frac 12\right)$. We establish exact asymptotics for the solution as both time and space variables tend to infinity and derive sharp growth rates for the H\"older coefficients. The proofs are based on Talagrand's majorizing measure theorem and Sudakov's minoration theorem.