Titres et résumés des exposés de WS2
- Yuri Bakhtin :
Invariant densities for dynamical systems with random switching
We consider a non-autonomous ordinary differential equation on a smooth manifold, with right-hand side that randomly switches between the elements of a finite family of smooth vector fields. For the resulting random dynamical system, we show that Hörmander-type hypoellipticity conditions are sufficient for uniqueness and absolute continuity of an invariant measure. Regularity properties of invariant densities will also be discussed. This is a joint work with Tobias Hurth.
- Jean-Baptiste Bardet : Rates of convergence in total variation norm for Markov processes
We present recent methods to obtain quantitative rates of convergence in total variation norm for Markov processes. We apply them to some examples of Piecewise Deterministic Markov Processes, in particular to the TCP window size process, which appears in the modeling of data transmission on the Internet. This is a joint work with A. Christen, A. Guillin, F. Malrieu and P.-A. Zitt.
- Bertrand Cloez : Exponential ergodicity for switching dynamical system
We study a class of Piecewise Deterministic Markov Processes with state space Rd \times E where E is a finite set. The continuous component evolves according to a smooth vector field that switches 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 a concrete criterion for the convergence to equilibrium in terms of Wasserstein distance. The proof is based on a coupling argument and a weak form of the Harris Theorem. In particular, we obtain exponential ergodicity in situations which do not verify any hypoellipticity assumption, but are not uniformly contracting either. We also obtain a bound in total variation distance under a suitable regularising assumption. This is a joint work with Martin Hairer.
- Francis Comets : Stochastic billiard in an inhomogeneous medium
In stochastic billiard, a particle is moving in straight line inside a domain in Euclidean space, and has random bounces upon hitting the boundary, according the cosine reflection law. We will start to review some of its main properties. To describe a large inhomgeneous medium, we then assume the domain is a random tube stretching in the first direction. We focus on the case with a drift: in this case, the jumps in the positive direction are always accepted while the jumps in the negative direction may be rejected. We will show that the motion is ballistic, finding a coupling with some random walk in a stationary ergodic random environment on Z with unbounded jumps.
- Marie Doumic : Nonparametric estimation of growth models: combining PDE, PDMP and statistics
Models describing the growth of populations have been developed based on assumptions on the stochastic mechanisms underlying growth and division at the single cell level. In particular, two different macroscopic models have been widely used for decades, assuming that cell division probability depends respectively on the age of the individuals (the renewal equation) or their size (the size-structured or growth-fragmentation equation) - or yet both. These PDE models may also be viewed as the Kolmogorov equation of the underlying PDMP process. We develop an estimation methodology to calibrate such models from experimental data, using either the PDMP or the PDE, prove error estimates, and test the hypothesis of an age-dependent versus size-dependent division rate. We also apply our method on real E. coli data. In collaboration with M. Hoffmann, N. Krell, P. Reynaud, L. Robert, V. Rivoirard.
- François Dufour : Optimal stopping for partially observed piecewise-deterministic Markov processes
This talk deals with the optimal stopping problem under partial observation for piecewise-deterministic Markov processes. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed optimal stopping problem. Then, we propose a numerical method, based on the quantization of the discrete-time filter process and the inter-jump times, to approximate the value function and to compute an $\epsilon$-optimal stopping time. We prove the convergence of the algorithms and bound the rates of convergence. This is a joint work with A. Brandejsky and B. de Saporta.
- Alessandra Faggionato :
Thermodynamics of Piecewise Deterministic Markov Processes
We discuss stationarity and time reversal symmetries for a large class of PDMPs. Increasing the frequency of the jumps, the system behaves asymptotically as deterministic and we investigate the structure of its fluctuations (i.e. deviations from the asymptotic behavior), recovering in a non Markovian frame a fluctuation-dissipation relation obtained by Bertini et. al. in the context of Markovian stochastic particle systems. Finally, motivated by our results for PDMPs we give a concise and universal solution of the variational problem introduced by M.D. Freidlin and A.D. Wentzell to characterize the static large deviation functional of a diffusion in the 1d torus. This is a joint work with D. Gabrielli and M. Ribezzi Crivellari.
- Joaquin Fontbona : Some bounds for the renewal theorem with spread-out inter arrival distributions
We will present some partial results aiming to establish quantitative exponential convergence estimates for the renewal theorem when the inter arrival distribution has a uniform component and one finite exponential monent. We improve to that end the classic idea of coupling the age processes of a delayed renewal process and a non delayed one in order to get explicit estimates. A key element is our construcion of a coalescent coupling between two copies of the process that succeeds with at least some explicit probability which is independent of their initial relative delay. This construction and the computation of the lower bound rely on the use of the uniform component to couple inter arrival lengths with small enough "shifts" of that random variables, combined in an optimized way with some concentration inequalities. A second element is the study of a biased random walk associated with two independent copies of the process, which allows us to obtain, via Lyapunov and martingale techniques, an estimate of the total elapsed time required to observe renewals of the two copies within a time-interval of prescribed length. If time permits, extensions of these ideas to PDMP will be discussed. Based on joint work in progress with Jean-Baptiste Bardet and Alejandra Christen.
- Hélène Guérin : Long time behavior of an ergodic variant of the telegraph process with jump rates depending on the position
We study a variant of the telegraph process in the sense that the jump rates of the velocity depends on the direction and on the position. Our aim is to control the total variation distance of this process to equilibrium at each instant. A path of a such process can be seen as the motion of a bacterium which is attracted by a nutriment with tumble phases depending on the distance between the bacterium and the nutriment. This is a joint work with F. Malrieu and J. Fontbona.
- Alexandre Genadot : Conductance Based Neuron Models
We consider a class of conductance based neuron models (CBNMs). These model consist in a deterministic dynamic coupled to a stochastic one. The deterministic behavior obeys a PDE which describes the propagation of the nerve impulse along a nerve fiber. This PDE is coupled to the dynamic of the ionic channels which is modelized by a jump markovian evolution. We present some recent results on CBNMs.
- Arnaud Guillin : Convergence de processus de Markov
Dans cet exposé, nous ferons un survol des différentes méthodes permettant d'étudier qualitativement ou quantitativement la vitesse de convergence vers l'équilibre de processus de Markov : méthode à la Meyn-Tweedie et fonctions de Lyapunov, inégalités fonctionnelles (Poincaré, Sobolev logarithmique,..), ou couplage (couplage synchrone, par réflexion,...).
- Aldéric Joulin : Intertwinings between Markov processes
Given a Markov process, we will see how a potential intertwining relation between some gradient and the underlying semigroup leads to long-time convergence results. Our approach covers diffusions as well as jump processes such as birth-death or TCP processes. The talk is based on two joint works with D. Chafai (Paris-Est Marne-la-Vallée) and M. Bonnefont (Bordeaux).
- Pierre Monmarché : The circular telegraph process
We study the relaxation to equilibrium of a toy model for non-symetric process, to investigate the interaction between the deterministic and stochastic parts of the dynamics. The observed oscillation in time of the L2 norm of the semi-group suggest a method relying at the end on a 3 order differential equation rather than, as usualy, a Gronwall Lemma.
- Philippe Robert : The Time Scales of a Stochastic Network with Failures
A simple transient Markov process with an absorbing point is used to investigate the qualitative behavior of a large scale storage network of non reliable file servers where files can be duplicated. When the size of the system goes to infinity it is shown that there is a critical value for the value of the number of files per server such that below this quantity, the system stays away from the absorbing state (all files lost) in a quasi-stationary state where most files have a maximum number of copies. Above this value, the network looses quickly a significant number of files until some equilibrium is reached. When the network is stable, it is shown that, with a convenient set of time scales and a representation of the associated Markov process as the solution of a Skorohod problem, the evolution of the network towards the absorbing state can be described in terms of a stochastic averaging principle. This is a joint work with Mathieu Feuillet.
- Florian Simatos : Analysis of a one-sided limit order book model
A limit order book is a financial trading mechanism that keeps track of orders made by traders, and allows to execute them in the future. In this talk I will present a simple model of a one-sided limit order book, which is modeled as a point process evolving over time. I will discuss two aspects of this model: the behavior of the so-called price process and its scaling limit. The proofs rely on a coupling with a branching random walk and on the theory of regenerative continuous trees. This is joint work (in progress) with Josh Reed (NYU).
- Maria Veretennikova : Continuous time random walks and fractional HJB equations
In this presentation you will be introduced to Continuous Time Random Walks which lead to fractional Hamilton Jacobi Bellman equations which have not been studied before. I will talk about the analysis of a simple version of the new fHJB and briefly discuss applications of this theory.
- Pierre-André Zitt : Large deviations for a gas of repulsing particles
We study a physical system of N interacting particles in R^3 subject to pair repulsion and confined by an external field. We will see that their empirical measure follows a large deviations principle as N tends to infinity.The rate function is minimized by a unique measure, which can be studied using ideas from Potential Theory. In the particular case of Coulomb interactions with a well chosen external field, this implies that the empirical distribution of the particles tends to an equilibrium measure which is uniform on a ball. This is a joint work with D. Chafaï and N. Gozlan.