Stochastic fractional heat equation with general rough noise

TL;DR

This paper studies the well-posedness of a nonlinear stochastic fractional heat equation driven by rough Gaussian noise, extending previous results to cases where the noise's Hurst index and fractional Laplacian order vary.

Contribution

It generalizes prior work by removing the zero condition on the coefficient function and provides new estimates for the fractional heat kernel.

Findings

Established well-posedness for the stochastic fractional heat equation with general rough noise.

Extended analysis to cases with fractional Laplacian order between 1 and 2.

Provided heat kernel estimates crucial for the analysis.

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.