Intermittency for the Hyperbolic Anderson Model with rough noise in space

TL;DR

This paper studies the stochastic wave equation driven by rough Gaussian noise with Hurst index between 1/4 and 1/2, proving existence, uniqueness, moment bounds, and weak intermittency of solutions.

Contribution

It establishes the existence and uniqueness of solutions in the Skorohod sense for the hyperbolic Anderson model with rough noise, extending previous work and analyzing intermittency.

Findings

Existence and uniqueness of solutions for H > 1/4

Exponential bounds on solution moments

Proof of weak intermittency of the solution

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.