Limit theorems for infinite-dimensional piecewise deterministic Markov processes. Applications to stochastic excitable membrane models

摘要

We present limit theorems for a sequence of Piecewise Deterministic MarkovProcesses (PDMPs) taking values in a separable Hilbert space. This class ofprocesses provides a rigorous framework for stochastic spatial models in whichdiscrete random events are globally coupled with continuous space-dependentvariables solving partial differential equations, e.g., stochastic hybridmodels of excitable membranes. We derive a law of large numbers whichestablishes a connection to deterministic macroscopic models and a martingalecentral limit theorem which connects the stochastic fluctuations to diffusionprocesses. As a prerequisite we carry out a thorough discussion of Hilbertspace valued martingales associated to the PDMPs. Furthermore, these limittheorems provide the basis for a general Langevin approximation to PDMPs, i.e.,stochastic partial differential equations that are expected to be similar intheir dynamics to PDMPs. We apply these results to compartmental-type models ofspatially extended excitable membranes. Ultimately this yields a system ofstochastic partial differential equations which models the internal noise of abiological excitable membrane based on a theoretical derivation from exactstochastic hybrid models.