Time-dependency in hyperbolic Anderson model: Stratonovich regime

Abstract

In this paper, the hyperbolic Anderson equation generated by a time-dependent Gaussian noise is under investigation in two fronts: The solvability and large-$t$ asymptotics. The investigation leads to a necessary and sufficient condition for existence and a precise large-$t$ limit form for the expectation of the solution. Three major developments are made for achieving these goals: A universal bound for Stratonovich moment that guarantees the Stratonovich integrability and ${\cal L}^2$-convergence of the Stratonovich chaos expansion under the best possible condition, a representation of the expected Stratonovich moments in terms of a time-randomized Brownian intersection local time, and a large deviation principle for the time-randomized Brownian intersection local time.