Construction of stochastic hybrid path integrals using operator methods

Stochastic hybrid systems involve the coupling between discrete and continuous stochastic processes. They are finding increasing applications in cell biology, ranging from modeling promoter noise gene networks to analyzing effects of stochastically-gated ion channels on voltage fluctuations single neurons neural networks. We have previously derived a path integral representation solutions associated differential Chapman-Kolmogorov equation, based representations Dirac delta function, used this determine ``least action'' paths noise-induced escape metastable state. In paper we present an alternative derivation integral, use bra-kets ``quantum-mechanical'' operators. show how operator method provides more efficient flexible framework for constructing integrals, which eliminates certain ad hoc steps previous context with regards general theory integrals. also highlight important role principal eigenvalues, spectral gaps Perron-Frobenius theorem. then perturbation methods develop various approximation schemes integrals moment generating functionals. First, consider Gaussian approximations loop expansions weak limit, analogous semi-classical limit quantum Second, identify analog weak-coupling by treating system as nonlinear Ornstein-Uhlenbeck process. This leads expansion moments terms products free propagators.