Generalized solutions to hyperbolic systems with random field coefficients

TL;DR

This paper develops a new mathematical framework using Colombeau algebras to analyze hyperbolic systems with random coefficients, enabling solutions under low regularity conditions not handled by classical stochastic methods.

Contribution

It introduces Colombeau algebra-based analysis for hyperbolic systems with irregular random coefficients, establishing existence, uniqueness, and connections to classical solutions.

Findings

Established existence and uniqueness of solutions in Colombeau framework

Provided new characterizations of Colombeau stochastic processes

Linked Colombeau solutions to classical weak solutions when available

Abstract

The paper addresses linear hyperbolic systems in one space dimension with random field coefficients. In many applications, a low degree of regularity of the paths of the coefficients is required, which is not covered by classical stochastic analysis. For this reason, we place our analysis in the framework of Colombeau algebras of generalized functions. We obtain new characterizations of Colombeau stochastic processes and establish existence and uniqueness of solutions in this framework. A number of applications to stochastic wave and transport equations are given and the Colombeau solutions are related to classical weak solutions, when the latter exist.