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We study the stochastic effect on the three-dimensional inviscid primitiveequations (PEs, also called the hydrostatic Euler equations). Specifically, weconsider a larger class of noises than multiplicative noises, and work in theanalytic function space due to the ill-posedness in Sobolev spaces of PEswithout horizontal viscosity. Under proper conditions, we prove the localexistence of martingale solutions and pathwise uniqueness. By adding verticalviscosity, i.e., considering the hydrostatic Navier-Stokes equations, we canrelax the restriction on initial conditions to be only analytic in thehorizontal variables with Sobolev regularity in the vertical variable, andallow the transport noise in the vertical direction. We establish the localexistence of martingale solutions and pathwise uniqueness, and show that thesolutions become analytic in the vertical variable instantaneously as $t>0$ andthe vertical analytic radius increases as long as the solutions exist.

Local Martingale Solutions and Pathwise Uniqueness for the Three-dimensional Stochastic Inviscid Primitive Equations