Well-posedness and large deviations for 3D primitive equations with jumps

We establish the global pathwise well-posedness and a Freidlin--Wentzell large deviation principle for the three-dimensional viscous primitive equations driven by multiplicative Lévy noise represented by a compensated Poisson random measure. Under local Lipschitz and moment growth assumptions on the noise coefficient, we construct local martingale solutions and obtain a unique maximal pathwise solution with càdlàg trajectories in the H¹ energy space. Additional sixth-moment growth in H¹ and L⁶ growth bounds on the baroclinic component of the noise yield global pathwise well-posedness. The proof combines the jump Itô formula with baroclinic and anisotropic energy estimates to rule out finite-time blow-up. Under further exponential integrability and structural assumptions on the noise coefficient, we establish a Freidlin--Wentzell large deviation principle in a strong path topology by the weak convergence approach.