Stochastic wave equation with additive fractional noise: solvability and global Hölder continuity

Shuhui Liu, Yaozhong Hu, Xiong Wang

Published 2023-05-03Version 1

We give the necessary and sufficient condition for the solvability of the stochastic wave equation $ \frac{\partial^2 }{\partial t^2}u(t,x) =\Delta u(t,x)+\dot{W}(t,x)$ on $\mathbb{R}^d$, where $W(t,x)$ is a fractional Brownian field with temporal Hurst parameter $H_0>1/2$ and spatial Hurst parameters $H_i\in(0,1)$ for $i=1,\cdots,d$. Moreover, we obtain the growth rate and the H\"older continuity of the solution on the whole space $\mathbb{R}^d$ when $d=1$ and $H_0=1/2$.