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