Stochastic Convective Wave Equation in Two Space Dimension

TL;DR

This paper investigates the stochastic convective wave equation in two dimensions driven by Gaussian noise, establishing existence, uniqueness, and regularity of solutions using Walsh's theory and Green's functions.

Contribution

It provides a necessary and sufficient condition for the existence of solutions and demonstrates their H"older continuity, advancing understanding of stochastic wave equations.

Findings

Existence and uniqueness of solutions under specific Gaussian noise conditions

H"older continuity of solutions proved using Green's function and Kolmogorov theorem

Necessary and sufficient condition for Gaussian noise source identified

Abstract

We study the convective wave equation in two space dimension driven by spatially homogeneous Gaussian noise. The existence of the real-valued solution is proved by providing a necessary and sufficient condition of Gaussian noise source. Our approach is based on the mild solution of the convective wave equation which is constructed by Walsh's theory of martingale measures. H\"older continuity of the solution is proved by using Green's function and Kolmogorov continuity theorem.