Construction of stochastic hybrid path integrals using operator methods

Abstract

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 operator methods, and show how this provides a more efficient and flexible framework for constructing hybrid path integrals in the weak noise limit. We also highlight the important role of principal eigenvalues, spectral gaps and the Perron–Frobenius theorem. Finally, we carry out a loop expansion of the associated moment generating functional in the weak noise limit, analogous to the semi-classical limit for quantum path integrals.