Stochastic fractional heat equation with general rough noise

Abstract

Consider the following nonlinear one-dimensional stochastic fractional heat equation $$\frac{\partial }{\partial t}u(t, x)= -(-\Delta)^{\alpha/2}u(t, x) +\sigma(t,x,u(t,x)) \dot{W}(t, x), $$ where $-(-\Delta)^{\alpha/2}$ is the fractional Laplacian on $\mathbb R$ for $1<\alpha<2$, and $\dot{W}$ is a Gaussian noise that is white in time and behaves in space as a fractional Brownian motion with Hurst index $H$ satisfying $\frac{3-\alpha}{4}<H<\frac12$. When $\alpha=2$, Hu and Wang ({\it Ann. Inst. Henri Poincar\'e Probab. Stat.} {\bf 58} (2022) 379-423) studied the well-posedness of the solution and its H\"older continuity, removing the technical condition $\sigma(0)=0$ that was previously assumed in Hu et al. ({\it Ann. Probab.} {\bf 45} (2017) 4561-4616). Their approach relied on working in a weighted space with a suitable power decay function. For the case $\alpha\in (1,2)$, inspired by Hu and Wang, we investigate the well-posedness of the stochastic fractional heat equation without imposing the technical condition of $\sigma(0)=0$, which was required in the earlier work of Liu and Mao ({\it Bull. Sci. Math.} {\bf181} (2022) 103207). In our analysis, precise estimates of the heat kernel associated with the fractional Laplacian $-(-\Delta)^{\alpha/2}$ play a crucial role.