On a class of stochastic hyperbolic equations with double characteristics

TL;DR

This paper investigates how Gaussian noise affects a class of hyperbolic PDEs with double characteristics, providing conditions for the existence of solutions based on spectral properties of the noise.

Contribution

It introduces explicit methods to analyze stochastic hyperbolic equations with polynomial coefficients and establishes conditions for solution existence considering colored spatial noise.

Findings

Derived explicit fundamental solutions for the PDE operator.

Established spectral conditions for the noise ensuring solution existence.

Analyzed the impact of polynomial coefficient dependence on solutions.

Abstract

We study the effect of Gaussian perturbations on a hyperbolic partial differential equation with double characteristics in two spatial dimensions. The coefficients of our partial differential operator depend polynomially on the space variables, while the noise is additive, white in time and coloured in space. We provide a sufficient condition on the spectral measure of the covariance functional describing the noise that allows for the existence of a random field solution for the resulting stochastic partial differential equation. Our approach is based on explicit computations for the fundamental solution of the partial differential operator and its Fourier transform.