Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Application to a stochastic Hodgkin−Huxley model^∗

¹ Johannes-Gutenberg-Universität Mainz, Institut für Mathematik, Staudingerweg 9, 55099 Mainz, Germany.
hoepfner@mathematik.uni-mainz.de
² Université de Cergy-Pontoise, Laboratoire AGM, UMR CNRS 8088, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France.
eva.loecherbach@u-cergy.fr
³ UniversitéPierre et Marie Curie, Laboratoire de Probabilités et Modèles Aléatoires, UMR CNRS 7599, Case 188, 4 place Jussieu, 75252 Paris cedex 5, France.
michele.thieullen@upmc.fr

Received: 5 March 2015
Revised: 7 April 2016
Accepted: 25 July 2016

Abstract

We formulate simple criteria for positive Harris recurrence of strongly degenerate stochastic differential equations with smooth coefficients on a state space with certain boundary conditions. The drift depends on time and space and is periodic in the time argument. There is no time dependence in the diffusion coefficient. Control systems play a key role, and we prove a new localized version of the support theorem. Beyond existence of some Lyapunov function, we only need one attainable inner point of full weak Hoermander dimension. Our motivation comes from a stochastic Hodgkin−Huxley model for a spiking neuron including its dendritic input. This input carries some deterministic periodic signal coded in its drift coefficient and is the only source of noise for the whole system. We have a 5d SDE driven by 1d Brownian motion. As an application of the general results above, we can prove positive Harris recurrence. Here analyticity of the coefficients and Nummelin splitting allow to formulate a Glivenko−Cantelli type theorem for the interspike intervals.