Intermittency for the Hyperbolic Anderson Model with rough noise in space

Abstract

In this article, we consider the stochastic wave equation on the real line driven by a linear multiplicative Gaussian noise, which is white in time and whose spatial correlation corresponds to that of a fractional Brownian motion with Hurst index $H\in (\frac14,\frac12)$. Initial data are assumed to be constant. First, we prove that this equation has a unique solution (in the Skorohod sense) and obtain an exponential upper bound for the $p$-th moment of the solution, for any $p\geq 2$. Condition $H>\frac14$ turns out to be necessary for the existence of solution. Secondly, we show that this solution coincides with the one obtained by the authors in a recent publication, in which the solution is interpreted in the It\^o sense. Finally, we prove that the solution of the equation in the Skorohod sense is weakly intermittent.