We discuss hypoelliptic and subelliptic diffusions; the lectures include the following topics: Malliavin calculus; Hormander's theorem; smoothness of transition probabilities under Hormander's brackets condition; control theory and Stroock-Varadhan's support theorems; hypoelliptic heat kernel estimates; gradient estimates and Harnack type inequalities for subelliptic diffusion semi-groups; notions of curvature related to sub-Riemannian diffusions.[-]

In this talk I will present a stochastic model for the excitability of a neuron in a network. The neuron described by an Hodgkin-Huxley type model receives from the network a random input which is a perturbation of a periodic deterministic signal. For such a model we study ergodicity properties. Then, we prove limit theorems in order to be able to estimate characteristics of the sequence of spiking times. This talk is based on a joint work with R. Hoepfner (Univ. Mainz) and E. Loecherbach (Univ. Cergy-Pontoise).

Hodgkin-Huxley model - ergodicity - limit theorems - estimation[-]

We study white noise perturbations of planar dynamical systems with heteroclinic networks in the limit of vanishing noise. We show that the probabilities of transitions between various cells that the network tessellates the plane into decay as powers of the noise magnitude, and we describe the underlying mechanism. A metastability picture emerges, with a hierarchy of time scales and clusters of accessibility, similar to the classical Freidlin-Wentzell picture but with shorter transition times. We discuss applications of our results to homogenization problems and to the invariant distribution asymptotics. At the core of our results are local limit theorems for exit distributions obtained via methods of Malliavin calculus. Joint work with Hong-Bin Chen and Zsolt Pajor-Gyulai.[-]