Local nondeterminism and local times of the stochastic wave equation driven by fractional-colored noise

TL;DR

This paper studies the local times of solutions to stochastic wave equations driven by fractional-colored Gaussian noise, establishing their existence, regularity, and path properties using Fourier analysis.

Contribution

It introduces new Fourier analytic techniques to prove local nondeterminism and the existence of continuous local times for fractional-colored noise driven wave equations.

Findings

Proved strong local nondeterminism of the solution.

Established existence of jointly continuous local times.

Analyzed differentiability and moduli of continuity of local times.

Abstract

We investigate the existence and regularity of the local times of the solution to a linear system of stochastic wave equations driven by a Gaussian noise that is fractional in time and colored in space. Using Fourier analytic methods, we establish strong local nondeterminism properties of the solution and the existence of jointly continuous local times. We also study the differentiability and moduli of continuity of the local times and deduce some sample path properties of the solution.