Transfer Operators for Stochastic Hybrid Systems on Manifolds with Guard-Induced Resets

This paper develops a transfer operator framework for stochastic hybrid systems with guard-induced resets, encompassing both the Koopman and Frobenius--Perron operators. Exploiting their duality, we derive a unified formulation in which observables and probability densities evolve under adjoint generators corresponding to the backward and forward Kolmogorov equations. The formulation is developed in a global and intrinsic manner on differentiable manifolds, ensuring consistency with the underlying geometric structure of the state space. In addition, we propose a finite volume computational scheme on manifolds that preserves total probability mass while accurately capturing fluxes across guards and reset-induced transfers. The proposed framework provides a unified and geometrically consistent approach to uncertainty propagation in stochastic hybrid systems, bridging continuous stochastic dynamics and hybrid transitions within a transfer operator perspective.