Time-dependency in hyperbolic Anderson model: Stratonovich regime

TL;DR

The paper analyzes the hyperbolic Anderson equation with time-dependent Gaussian noise in the Stratonovich sense, establishing a sharp solvability criterion via a finite integrability condition on the spectral measure and proving $L^2$-convergence of the Stratonovich chaos expansion. It then represents Stratonovich moments through a time-randomized Brownian intersection local time and proves a large deviation principle for this object, enabling precise large-time asymptotics for the first moment of the solution. The main result delivers a rigorous, fully explicit long-time scaling for ${\mathbb E}[u(t,x)]$ under the regime $\alpha_0+\alpha<2$, showing that ${\mathbb E}[u(t,x)]$ grows like $\exp\{ t^{\frac{4-\alpha-\alpha_0}{3-\alpha}} \times C(\alpha,\alpha_0,\mathcal M) \}$ up to subexponential factors, with $\mathcal M$ a variational constant capturing intermittency. The methods blend Stratonovich chaos analysis, Green's function techniques for the wave equation, and Brownian-time randomization to reveal how time-dependence in the noise shapes solvability and intermittency, providing sharp criteria and asymptotics with potential implications for related SPDEs and intermittent phenomena.

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.

Figures (1)

• Figure 1: triangular approximation

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• Theorem 3.1
• Theorem 4.1