Growth rates for the Hölder coefficients of the linear stochastic fractional heat equation with rough dependence in space

Abstract: We study the linear stochastic fractional heat equation $$ \frac{\partial}{\partial t}u(t,x)=-(-\Delta)^{\frac{\alpha}{2}}u(t,x)+\dot{W}(t,x), \quad t>0, \quad x\in\mathbb{R}, $$ 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 that is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in\left(\frac{2-\alpha}{2},\frac{1}{2}\right)$. We establish exact asymptotics for the solution as $t, x \to \infty$ 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.