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

TL;DR

This paper introduces a geometric transfer operator framework for stochastic hybrid systems on manifolds, enabling consistent uncertainty propagation across hybrid transitions and guard-induced resets.

Contribution

It develops a unified, intrinsic formulation of transfer operators on manifolds, incorporating guard resets and a finite volume scheme that preserves probability mass.

Findings

Framework unifies Koopman and Frobenius--Perron operators for hybrid systems.

Finite volume scheme accurately captures fluxes across resets.

Ensures geometric consistency and probability mass preservation.

Abstract

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.