Stochastic wave equation with additive fractional noise: solvability and global H\"older continuity

Abstract

We determine the range of Hurst parameters that provide the necessary and sufficient conditions for the solvability, in $L^2(\Omega)$, of the stochastic wave equation: $ \frac{\partial^2 }{\partial t^2}u(t,x) =\Delta u(t,x)+\dot{W}(t,x)$, where $\{ W(t,x),\ t\geq 0, x\in \mathbb{R}^d\} $ is a fractional Brownian field with temporal Hurst parameter $H_0\in[\tfrac12,1]$ and spatial Hurst parameters $H_i\in(0,1)$ for $i=1,\cdots,d$. {In particular, the solvability condition exhibits a phase transition at $H_0 = 1$.} We also obtain the sharp growth rate and the sharp H\"older continuity of the solution on the real line in the case $H_0=1/2$.