  We study the stochastic effect on the three-dimensional inviscid primitive
equations (PEs, also called the hydrostatic Euler equations). Specifically, we
consider a larger class of noises than multiplicative noises, and work in the
analytic function space due to the ill-posedness in Sobolev spaces of PEs
without horizontal viscosity. Under proper conditions, we prove the local
existence of martingale solutions and pathwise uniqueness. By adding vertical
viscosity, i.e., considering the hydrostatic Navier-Stokes equations, we can
relax the restriction on initial conditions to be only analytic in the
horizontal variables with Sobolev regularity in the vertical variable, and
allow the transport noise in the vertical direction. We establish the local
existence of martingale solutions and pathwise uniqueness, and show that the
solutions become analytic in the vertical variable instantaneously as $t>0$ and
the vertical analytic radius increases as long as the solutions exist.
