Stochastic wave equation with additive fractional noise: solvability and global Hölder 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 $\{W(t,x),t\ge 0,x\in {\mathbb{R}}^d\}$ is a fractional Brownian field with temporal Hurst parameter $H_0\in [\frac{1}{2},1]$ 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 Hölder continuity of the solution on the real line in the case $H_0=1/2$.

arXiv comment35 pages, 3 figures