Pathwise Solutions for Stochastic Hydrostatic Euler Equations under the Local Rayleigh Condition

The hydrostatic Euler equations are important in the study of atmospheric and oceanic dynamics in the planetary scale. While its deterministic version has been widely studied in the literature, its stochastic version is far less understood. In this paper, we consider the two-dimensional stochastic hydrostatic Euler equations with initial data that are random variables in a suitable Sobolev space satisfying the local Rayleigh condition. We establish local-in-time existence and uniqueness of maximal pathwise solutions. Our work provides the first result on existence and uniqueness in Sobolev spaces, and establishes the first existence of pathwise solutions to the stochastic hydrostatic Euler equations.