  The deterministic inviscid primitive equations (also called the hydrostatic
Euler equations) are known to be ill-posed in Sobolev spaces and in Gevrey
classes of order strictly greater than 1, and some of their analytic solutions
exist only locally in time and exhibit finite-time blowup. This work
demonstrates that introducing suitable random noise can restore the local
well-posedness and prevent finite-time blowups. Specifically, random diffusion
addresses the ill-posedness in certain Gevrey classes, allowing us to establish
the local well-posedness almost surely and the global existence of solutions
with high probability. In the case of random damping (linear multiplicative
noise), the noise prevents analytic solutions from forming singularities in
finite time, resulting in globally existing solutions with high probability.
