  We present limit theorems for a sequence of Piecewise Deterministic Markov
Processes (PDMPs) taking values in a separable Hilbert space. This class of
processes provides a rigorous framework for stochastic spatial models in which
discrete random events are globally coupled with continuous space-dependent
variables solving partial differential equations, e.g., stochastic hybrid
models of excitable membranes. We derive a law of large numbers which
establishes a connection to deterministic macroscopic models and a martingale
central limit theorem which connects the stochastic fluctuations to diffusion
processes. As a prerequisite we carry out a thorough discussion of Hilbert
space valued martingales associated to the PDMPs. Furthermore, these limit
theorems provide the basis for a general Langevin approximation to PDMPs, i.e.,
stochastic partial differential equations that are expected to be similar in
their dynamics to PDMPs. We apply these results to compartmental-type models of
spatially extended excitable membranes. Ultimately this yields a system of
stochastic partial differential equations which models the internal noise of a
biological excitable membrane based on a theoretical derivation from exact
stochastic hybrid models.
