Wave equation with a coloured stable noise

TL;DR

This paper constructs a new type of random measure driven by a stable fractional field and studies the regularity of solutions to a wave equation with this stochastic source, revealing conditions for continuity and absolute continuity.

Contribution

It introduces a novel stable fractional noise-based measure and analyzes the regularity of wave equation solutions driven by this measure.

Findings

The measure is σ-additive in probability.

Solutions are Hölder continuous up to a certain order.

Solutions are absolutely continuous for Hurst index in (2/3,1).

Abstract

We define a random measure generated by a real anisotropic harmonizable fractional stable field $Z^H$ with stability parameter $\alpha\in(1,2)$ and Hurst index $H\in(1/2,1)$ and prove that the measure is $\sigma$-additive in probability. An integral with respect to this measure is constructed, which enables us to consider a wave equation in $\mathbb R^3$ with a random source generated by $Z^H$. We show that the solution to this equation, given by Kirchhoff's formula, has a modification, which is H\"older continuous of any order up to $(3H-1)\wedge 1$. In the case where $H\in(2/3,1)$, we show further that the modification is absolutely continuous.