In the present work we derive a Central Limit Theorem for sequences of Hilbert-valued Piecewise Deterministic Markov process models and their global fluctuations around their deterministic limit identified by the Law of Large Numbers. We provide a version of the limiting fluctuations processes in the form of a distribution valued stochastic partial differential equation which can be the starting point for further theoretical and numerical analysis. We also present applications of our results to two examples of hybrid models of spatially extended excitable membranes: compartmental-type neuron models and neural fields models. These models are fundamental in neuroscience modelling both for theory and numerics.Keywords: Piecewise Deterministic Markov Processes; infinite-dimensional stochastic processes; law of large numbers; central limit theorem; global fluctuations; Langevin approximation; stochastic excitable media; neuronal membrane models; neural field models MSC 2010: 60B12; 60F05; 60J25; 92C20; * This work has been supported by the Agence Nationale de la Recherche through the ANR Project MANDy "Mathematical Analysis of Neuronal Dynamics" ANR-09-BLAN-0008.
Spatio-Temporal Piecewise Deterministic ProcessesWe give a brief definition of hybrid models / PDMPs relevant for the present study and refer to [10,17,25] for a more detailed discussion. Let (Ω, F , (F t ) t≥0 , P) be a filtered probability space satisfying the usual conditions, X and H denote separable Hilbert spaces forming an evolution triplet X ⊂ H ⊂ X * and K be an at most countable set. In this study all spaces and sets are equipped with their Borel-σ-fields and measurability always means Borel-measurability. Then a PDMP (U t , Θ t ) t≥0 is a càdlàg strong Markov process taking values in H × K which is uniquely defined by the quadruple (A, B, Λ, µ) in the following sense:(i) The operators A : X × K → X * , which is linear in its X-argument, and B : X × K → X * are such that the abstract evolution equationṡare well-posed in the weak sense for any initial condition u 0 ∈ H, i.e., there exists a unique weak solution u to (2.1) such that u ∈ L 2 ((0, T ),We denote by φ t (u 0 , θ) the value of the unique solution corresponding to the parameter θ at time t started in u 0 . Then the PDMP (U t , Θ t ) t≥0 satisfies U t = φ t−τ k (U τ k ) for t ∈ [τ k , τ k+1 ), where the random variables τ k , k ∈ N, denote the jump-times of the PDMP with τ 0 = 0.