9 research outputs found
Quantitative speeds of convergence for exposure to food contaminants
In this paper we consider a class of piecewise-deterministic Markov processes
(PDMPs) modeling the quantity of a given food contaminant in the body. On the
one hand, the amount of contaminant increases with random food intakes and, on
the other hand, decreases thanks to the release rate of the body. Our aim is to
provide quantitative speeds of convergence to equilibrium for the total
variation and Wasserstein distances via coupling methods.Comment: 20 page
Fluctuations of the Empirical Measure of Freezing Markov Chains
In this work, we consider a finite-state inhomogeneous-time Markov chain
whose probabilities of transition from one state to another tend to decrease
over time. This can be seen as a cooling of the dynamics of an underlying
Markov chain. We are interested in the long time behavior of the empirical
measure of this freezing Markov chain. Some recent papers provide almost sure
convergence and convergence in distribution in the case of the freezing speed
, with different limits depending on or
. Using stochastic approximation techniques, we generalize these
results for any freezing speed, and we obtain a better characterization of the
limit distribution as well as rates of convergence as well as functional
convergence.Comment: 30 page
Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories
In this work, we consider an inhomogeneous (discrete time) Markov chain and
are interested in its long time behavior. We provide sufficient conditions to
ensure that some of its asymptotic properties can be related to the ones of a
homogeneous (continuous time) Markov process. Renowned examples such as a
bandit algorithms, weighted random walks or decreasing step Euler schemes are
included in our framework. Our results are related to functional limit
theorems, but the approach differs from the standard "Tightness/Identification"
argument; our method is unified and based on the notion of pseudotrajectories
on the space of probability measures
Étude quantitative de processus de Markov déterministes par morceaux issus de la modélisation
The purpose of this Ph.D. thesis is the study of piecewise deterministic Markov processes, which are often used for modeling many natural phenomena. Precisely, we shall focus on their long time behavior as well as their speed of convergence to equilibrium, whenever they possess a stationary probability measure. Providing sharp quantitative bounds for this speed of convergence is one of the main orientations of this manuscript, which will usually be done through coupling methods. We shall emphasize the link between Markov processes and mathematical fields of research where they may be of interest, such as partial differential equations. The last chapter of this thesis is devoted to the introduction of a unified approach to study the long time behavior of inhomogeneous Markov chains, which can provide functional limit theorems with the help of asymptotic pseudotrajectories.L'objet de cette thèse est d'étudier une certaine classe de processus de Markov, dits déterministes par morceaux, ayant de très nombreuses applications en modélisation. Plus précisément, nous nous intéresserons à leur comportement en temps long et à leur vitesse de convergence à l'équilibre lorsqu'ils admettent une mesure de probabilité stationnaire. L'un des axes principaux de ce manuscrit de thèse est l'obtention de bornes quantitatives fines sur cette vitesse, obtenues principalement à l'aide de méthodes de couplage. Le lien sera régulièrement fait avec d'autres domaines des mathématiques dans lesquels l'étude de ces processus est utile, comme les équations aux dérivées partielles. Le dernier chapitre de cette thèse est consacré à l'introduction d'une approche unifiée fournissant des théorèmes limites fonctionnels pour étudier le comportement en temps long de chaînes de Markov inhomogènes, à l'aide de la notion de pseudo-trajectoire asymptotique
LONG TIME BEHAVIOR OF MARKOV PROCESSES AND BEYOND
International audienceThis note provides several recent progresses in the study of long time behavior of Markov processes. The examples presented below are related to other scientific fields as PDE's, physics or biology. The involved mathematical tools as propagation of chaos, coupling, functional inequalities, provide a good picture of the classical methods that furnish quantitative rates of convergence to equilibrium.Cet article présente plusieurs progrés récents dans l'étude du comportement en temps long de certains processus de Markov. Les exemples présentés ci-dessous sont motivés par différentes applications issues de la physique ou de la biologie. Les outils mathématiques employés, propagation du chaos, couplage, inégalités fonctionnelles, couvrent un large spectre des techniques disponibles pour obtenir des comportements en temps long quantitatifs
Quantitative study of piecewise deterministic Markov processes arising in modelization
L'objet de cette thèse est d'étudier une certaine classe de processus de Markov, dits déterministes par morceaux, ayant de très nombreuses applications en modélisation. Plus précisément, nous nous intéresserons à leur comportement en temps long et à leur vitesse de convergence à l'équilibre lorsqu'ils admettent une mesure de probabilité stationnaire. L'un des axes principaux de ce manuscrit de thèse est l'obtention de bornes quantitatives fines sur cette vitesse, obtenues principalement à l'aide de méthodes de couplage. Le lien sera régulièrement fait avec d'autres domaines des mathématiques dans lesquels l'étude de ces processus est utile, comme les équations aux dérivées partielles. Le dernier chapitre de cette thèse est consacré à l'introduction d'une approche unifiée fournissant des théorèmes limites fonctionnels pour étudier le comportement en temps long de chaînes de Markov inhomogènes, à l'aide de la notion de pseudo-trajectoire asymptotique.The purpose of this Ph.D. thesis is the study of piecewise deterministic Markov processes, which are often used for modeling many natural phenomena. Precisely, we shall focus on their long time behavior as well as their speed of convergence to equilibrium, whenever they possess a stationary probability measure. Providing sharp quantitative bounds for this speed of convergence is one of the main orientations of this manuscript, which will usually be done through coupling methods. We shall emphasize the link between Markov processes and mathematical fields of research where they may be of interest, such as partial differential equations. The last chapter of this thesis is devoted to the introduction of a unified approach to study the long time behavior of inhomogeneous Markov chains, which can provide functional limit theorems with the help of asymptotic pseudotrajectories
Long time behavior of Markov processes and beyond*
This note provides several recent progresses in the study of long time behavior of Markov
processes. The examples presented below are related to other scientific fields as PDE’s,
physics or biology. The involved mathematical tools as propagation of chaos, coupling,
functional inequalities, provide a good picture of the classical methods that furnish
quantitative rates of convergence to equilibrium