On some stochastic hyperbolic equations with symplectic characteristics

TL;DR

This paper investigates the impact of Gaussian noise on hyperbolic PDEs with symplectic structures, establishing conditions for solutions and analyzing how symplectic geometry influences noise regularity requirements.

Contribution

It extends existing work by providing explicit conditions for the existence of solutions to stochastic hyperbolic equations with symplectic features and analyzes the influence of symplectic geometry on noise regularity.

Findings

Derived spectral measure conditions for solution existence

Analyzed the effect of symplectic structure on noise regularity

Provided explicit fundamental solutions and Fourier transforms

Abstract

We study the effect of Gaussian perturbations on a class of model hyperbolic partial differential equations with double symplectic characteristics in low spatial dimensions, extending some recent work in [5]. The coefficients of our partial differential operators contain harmonic oscillators in the space variables, while the noise is additive, white in time and colored in space. We provide sufficient conditions 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. Furthermore we show how the symplectic structure of the set of multiple points affects the regularity of the noise needed to build a measurable process solution. Our approach is based on some explicit computations for the fundamental solutions of several model partial differential operators together with their explicit Fourier transforms.