Wiener integrals with respect to the Hermite random field and applications to the wave equation

TL;DR

This paper introduces Wiener integrals with respect to the Hermite random field, explores their properties, and applies them to analyze the wave equation driven by this field, including existence, regularity, and density of solutions.

Contribution

It defines Wiener integrals for the Hermite random field and applies them to study the wave equation driven by this process, establishing existence and regularity results.

Findings

Existence of solutions to the wave equation driven by the Hermite sheet

Regularity properties of the solution's sample paths

Existence of the density and local times of the solution

Abstract

The Hermite random field has been introduced as a limit of some weighted Hermite variations of the fractional Brownian sheet. In this work we define it as a multiple integral with respect to the standard Brownian sheet and introduce Wiener integrals with respect to it. As an application we study the wave equation driven by the Hermite sheet. We prove the existence of the solution and we study the regularity of its sample paths, the existence of the density and of its local times. 2000 AMS Classification Numbers: 60F05, 60H05, 60G18.