860 research outputs found
Some simple but challenging Markov processes
In this note, we present few examples of Piecewise Deterministic Markov
Processes and their long time behavior. They share two important features: they
are related to concrete models (in biology, networks, chemistry,. . .) and they
are mathematically rich. Their math-ematical study relies on coupling method,
spectral decomposition, PDE technics, functional inequalities. We also relate
these simple examples to recent and open problems
Propagation of chaos and Poincar\'e inequalities for a system of particles interacting through their cdf
In the particular case of a concave flux function, we are interested in the
long time behaviour of the nonlinear process associated to the one-dimensional
viscous scalar conservation law. We also consider the particle system obtained
by remplacing the cumulative distribution function in the drift coefficient of
this nonlinear process by the empirical cdf. We first obtain trajectorial
propagation of chaos result. Then, Poincar\'e inequalities are used to get
explicit estimates concerning the long time behaviour of both the nonlinear
process and the particle system
Lotka-Volterra with randomly fluctuating environments: a full description
In this note, we study the long time behavior of Lotka-Volterra systems whose
coefficients vary randomly. Benam and Lobry established that randomly switching
between two environments that are both favorable to the same species may lead
to four different regimes: almost sure extinction of one of the two species,
random extinction of one species or the other and persistence of both species.
Our purpose here is to provide a complete description of the model. In
particular, we show that any couple of environments may lead to the four
different behaviours of the stochastic process depending on the jump rates
Quantitative estimates for the long time behavior of an ergodic variant of the telegraph process
Motivated by stability questions on piecewise deterministic Markov models of
bacterial chemotaxis, we study the long time behavior of a variant of the
classic telegraph process having a non-constant jump rate that induces a drift
towards the origin. We compute its invariant law and show exponential
ergodicity, obtaining a quantitative control of the total variation distance to
equilibrium at each instant of time. These results rely on an exact description
of the excursions of the process away from the origin and on the explicit
construction of an original coalescent coupling for both velocity and position.
Sharpness of the obtained convergence rate is discussed.Comment: Definitive version of former paper "Quantitative estimates for the
long time behavior of a PDMP describing the movement of bacteria", now
accepted in Advances in Applied Probability. Presentation changed. A
diffusive scaling limit result is added. Sharpness of the long-time
convergence rate is discussed. 20 pages, 3 figure
Long time behavior of telegraph processes under convex potentials
We study the long-time behavior of variants of the telegraph process with
position-dependent jump-rates, which result in a monotone gradient-like drift
toward the origin. We compute their invariant laws and obtain, via
probabilistic couplings arguments, some quantitative estimates of the total
variation distance to equilibrium. Our techniques extend ideas previously
developed for a simplified piecewise deterministic Markov model of bacterial
chemotaxis.Comment: 26 pages, 3 figure
Probabilistic approach for granular media equations in the non uniformly convex case
We use here a particle system to prove a convergence result as well as a
deviation inequality for solutions of granular media equation when the
confinement potential and the interaction potential are no more uniformly
convex. Proof is straightforward, simplifying deeply proofs of
Carrillo-McCann-Villani \cite{CMV,CMV2} and completing results of Malrieu
\cite{malrieu03} in the uniformly convex case. It relies on an uniform
propagation of chaos property and a direct control in Wasserstein distance of
solutions starting with different initial measures. The deviation inequality is
obtained via a transportation cost inequality replacing the logarithmic
Sobolev inequality which is no more clearly dimension free
Long time behavior of diffusions with Markov switching
Let be an Ornstein-Uhlenbeck diffusion governed by an ergodic finite
state Markov process : ,
given. Under ergodicity condition, we get quantitative estimates for the long
time behavior of . We also establish a trichotomy for the tail of the
stationary distribution of : it can be heavy (only some moments are finite),
exponential-like (only some exponential moments are finite) or Gaussian-like
(its Laplace transform is bounded below and above by Gaussian ones). The
critical moments are characterized by the parameters of the model
On the Laplace transform of perpetuities with thin tails
We consider the random variables which are solutions of the
distributional equation R\overset{\cL}{=}MR+Q, where is independent
of and \ABS{M}\leq 1. Goldie and Gr\"ubel showed that the tails of
are no heavier than exponential. In this note we provide the exact lower and
upper bounds of the domain of the Laplace transform of
Les images chez les enfants et chez les primitifs
International audienceLa notion d'une pensée prélogique, commune aux enfants et aux primitifs, a contribué à nous délivrer de la croyance en la permanence des fonctions. Les correspondances que l'on croyait pouvoir affirmer entre la magie spontanée des enfants et les coutumes propitiatoires des primitifs, entres les interprétation animistes ou artificialistes des uns et des autres, nous ont familiarisé avec l'idée d'une histoire de l'esprit
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