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We establish the existence and uniqueness of pathwise strong solutions to thestochastic 3D primitive equations with only horizontal viscosity anddiffusivity driven by transport noise on a cylindrical domain $M=(-h,0) \timesG$, $G\subset \mathbb{R}^2$ bounded and smooth, with the physical Dirichletboundary conditions on the lateral part of the boundary. Compared to thedeterministic case where the uniqueness of $z$-weak solutions holds in $L^2$,more regular initial data are necessary to establish uniqueness in theanisotropic space $H^1_z L^2_{xy}$ so that the existence of local pathwisesolutions can be deduced from the Gy\"{o}ngy-Krylov theorem. Global existenceis established using the logarithmic Sobolev embedding, the stochastic Gronwalllemma and an iterated stopping time argument.

Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity