Growth rates for the Hölder coefficients of the linear stochastic fractional heat equation with rough dependence in space
Chang Liu, Bin Qian, Ran Wang
TL;DR
The paper analyzes the linear stochastic fractional heat equation with a rough spatial Gaussian noise, deriving exact large-time and large-space asymptotics for the solution and sharp growth rates for Hölder coefficients. By representing the solution with the fractional Green kernel and exploiting Gaussian process tools, it establishes precise domain-size and time-space scaling through the scaling function $\Psi(t,L)$ and $\sqrt{\log}$-type corrections. The main contributions are (i) exact asymptotics for $\mathbb{E}\big[\sup_{0\le t\le T,|x|\le L}u(t,x)\big]$, (ii) sharp Hölder regularity for spatial and temporal increments with explicit exponents, and (iii) a unified treatment of domain-size effects under spatial roughness, extending results from the white-noise case to rough noise via majorizing measure and minoration techniques. These results enhance understanding of nonlinear SPDEs driven by rough noise by providing precise decay- and log-growth controls essential for stability and well-posedness analyses of related nonlinear models.
Abstract
We study the linear stochastic fractional heat equation $$ \frac{\partial}{\partial t}u(t,x)=-(-Δ)^{\fracα2}u (t,x)+\dot{W}(t,x),\ \ t> 0,\ \ x\in\RR, $$ 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 $H\in\left(\frac {2-α}2,\frac 12\right)$. We establish exact asymptotics for the solution as both time and space variables tend to infinity and derive sharp growth rates for the Hölder coefficients. The proofs are based on Talagrand's majorizing measure theorem and Sudakov's minoration theorem.