Stéphane Le Borgne

Qualitative properties of certain piecewise deterministic Markov processes

We study a class of Piecewise Deterministic Markov Processes with state space Rd x E where E is a finite set. The continuous component evolves according to a smooth vector field that is switched at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Working under the general assumption that the process stays in a compact set, we detail a possible construction of the process and characterize its support, in terms of the solutions set of a differential inclusion. We establish results on the long time behaviour of the process, in relation to a certain set of accessible points, which is shown to be strongly linked to the support of invariant measures. Under Hormander-type bracket conditions, we prove that there exists a unique invariant measure and that the processes converges to equilibrium in total variation. Finally we give examples where the bracket condition does not hold, and where there may be one or many invariant measures, depending on the jump rates between the flows.

Quantitative ergodicity for some switched dynamical systems

We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space

Limit theorems for non-hyperbolic automorphisms of the torus

\mathbb{R}^d\times E

On the stability of planar randomly switched systems

where

Limit law for some modified ergodic sums

E

Martingales in Hyperbolic Geometry

is a finite set. The continous component evolves according to a smooth vector field that it switched at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances.

Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques

We prove the Donsker and Strassen invariance principles and other results for ergodic sums associated to regular functions for non-hyperbolic automorphisms of the torus. For this we use arithmetical and geometrical considerations which allow us to apply Gordin’s method.

Principes d'invariance pour les flots diagonaux sur SL(d,R)/SL(d,Z)

Consider the random process (Xt) t>0 solution of u Xt = AItXt where (It) t>0 is a Markov process on f 0;1g and A0 and A1 are real Hurwitz matrices on R 2 . Assuming that there exists � 2 (0;1) such that (1 − � )A0 + �A 1 has a positive eigenvalue, we establish that k Xtk may converge to 0 or +1 depending on the the jump rate of the process I. An application to product of random matrices is studied. This paper can be viewed as a probabilistic counterpart of the paper [2] by Balde, Boscain and Mason.

Théorème limite central presque sûr pour les marches aléatoires avec trou spectral Quenched central limit theorem for random walks with a spectral gap

An example due to Erdos and Fortet shows that, for a lacunary sequence of integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a generalization of their example interpreted as the limiting behavior of some modified ergodic sums in the framework of dynamical systems.

Théorème limite central presque sûr pour les marches aléatoires avec trou spectral

The famous De Moivre-Laplace theorem states the convergence toward a gaussian law of \(\sum\limits_{j=0}^{n-1}Y _{j}/\sqrt{n}\) when the Y i are independent, centered, identically distributed random variables in L 2. This result is usually named Central Limit Theorem (CLT). The convergence still holds in some non independent cases (Markov chains, α- or ϕ-mixing processes, martingales,…). Here we are interested in stationary processes defined by regular functions on regular hyperbolic systems. We show how the martingales formalism is well fitted to get the CLT in such situations. First we prove a few results on martingales and present Gordin’s method. Then we employ the method for two toy model dynamical systems: the angle doubling on the circle and the cat map. After what, the presented ideas are applied to more general dynamical systems (among which 1960 Sinai’s example and some other geodesic flows on hyperbolic manifolds). We stress the importance of the equirepartition of some submanifolds and explain how this can be related to the mixing properties of the system. As an example of application, we study certain asymptotic properties of random walks on \(\mathbb{R}^{d}\) driven by a hyperbolic system.