Academic literature on the topic 'Invariant distribution of Markov processes'

Cette thèse étudie le comportement en temps long des particules run-and-tumble (RTPs), un modèle pour les bactéries en physique statistique hors équilibre, en utilisant des processus de Markov déterministes par morceaux (PDMPs). La motivation est d'améliorer la compréhension au niveau particulaire des phénomènes actifs, en particulier la séparation de phase induite par la motilité (MIPS). La mesure invariante pour deux RTPs avec jamming sur un tore 1D est déterminée pour mécanismes de tumble et jamming généraux, révélant deux classes d'universalité hors équilibre. De plus, la dépendance du temps de mélange en fonction des paramètres du modèle est déterminée en utilisant des techniques de couplage et le modèle continu PDMP est rigoureusement relié à un modèle sur réseau connu. Dans le cas de deux RTPs avec jamming sur la droite réelle et interagissant à travers un potentiel attractif, la mesure invariante présente des différences qualitatives en fonction des paramètres du modèle, rappelant des transitions de forme et des classes d'universalité. Des taux de convergence fins sont à nouveau obtenus par des techniques de couplage. Par ailleurs, la mesure invariante explicite de trois RTPs se bloquant sur le tore 1D est calculée. Enfin, les résultats de convergence hypocoercive sont étendus aux RTPs, obtenant ainsi des taux de convergence \( L^2 \) fins dans un cadre général qui couvre également les PDMPs utilisés pour l'échantillonnage et Langevin cinétique<br>This thesis investigates the long-time behavior of run-and-tumble particles (RTPs), a model for bacteria's moves and interactions in out-of-equilibrium statistical mechanics, using piecewise deterministic Markov processes (PDMPs). The motivation is to improve the particle-level understanding of active phenomena, in particular motility induced phase separation (MIPS). The invariant measure for two jamming RTPs on a 1D torus is determined for general tumbling and jamming, revealing two out-of-equilibrium universality classes. Furthermore, the dependence of the mixing time on model parameters is established using coupling techniques and the continuous PDMP model is rigorously linked to a known on-lattice model. In the case of two jamming RTPs on the real line interacting through an attractive potential, the invariant measure displays qualitative differences based on model parameters, reminiscent of shape transitions and universality classes. Sharp quantitative convergence bounds are again obtained through coupling techniques. Additionally, the explicit invariant measure of three jamming RTPs on the 1D torus is computed. Finally, hypocoercive convergence results are extended to RTPs, achieving sharp \( L^2 \) convergence rates in a general setting that also covers kinetic Langevin and sampling PDMPs

La première partie de cette thèse porte sur les automates cellulaires probabilistes (ACP) sur la ligne et à deux voisins. Pour un ACP donné, nous cherchons l'ensemble de ces lois invariantes. Pour des raisons expliquées en détail dans la thèse, ceci est à l'heure actuelle inenvisageable de toutes les obtenir et nous nous concentrons, dans cette thèse, surles lois invariantes markoviennes. Nous établissons, tout d'abord, un théorème de nature algébrique qui donne des conditions nécessaires et suffisantes pour qu'un ACP admette une ou plusieurs lois invariantes markoviennes dans le cas où l'alphabet E est fini. Par la suite, nous généralisons ce résultat au cas d'un alphabet E polonais après avoir clarifié les difficultés topologiques rencontrées. Enfin, nous calculons la fonction de corrélation du modèleà 8 sommets pour certaines valeurs des paramètres du modèle en utilisant une partie desrésultats précédents<br>The first part of this thesis is about probabilistic cellular automata (PCA) on the line and with two neighbors. For a given PCA, we look for the set of its invariant distributions. Due to reasons explained in detail in this thesis, it is nowadays unthinkable to get all of them and we concentrate our reections on the invariant Markovian distributions. We establish, first, an algebraic theorem that gives a necessary and sufficient condition for a PCA to have one or more invariant Markovian distributions when the alphabet E is finite. Then, we generalize this result to the case of a polish alphabet E once we have clarified the encountered topological difficulties. Finally, we calculate the 8-vertex model's correlation function for some parameters values using previous results.The second part of this thesis is about infinite iterations of stochastic processes. We establish the convergence of the finite dimensional distributions of the α-stable processes iterated n times, when n goes to infinite, according to parameter of stability and to drift r. Then, we describe the limit distributions. In the iterated Brownian motion case, we show that the limit distributions are linked with iterated functions system

We studied invariant measures and invariant densities for dynamical systems with random switching (switching systems, in short). These switching systems can be described by a two-component Markov process whose first component is a stochastic process on a finite-dimensional smooth manifold and whose second component is a stochastic process on a finite collection of smooth vector fields that are defined on the manifold. We identified sufficient conditions for uniqueness and absolute continuity of the invariant measure associated to this Markov process. These conditions consist of a Hoermander-type hypoellipticity condition and a recurrence condition. In the case where the manifold is the real line or a subset of the real line, we studied regularity properties of the invariant densities of absolutely continuous invariant measures. We showed that invariant densities are smooth away from critical points of the vector fields. Assuming in addition that the vector fields are analytic, we derived the asymptotically dominant term for invariant densities at critical points.

A Hidden Markov Model generates two basic stochastic processes, a Markov chain, which is hidden, and an observation sequence. The filtering process of a Hidden Markov Model is, roughly speaking, the sequence of conditional distributions of the hidden Markov chain that is obtained as new observations are received. It is well-known, that the filtering process itself, is also a Markov chain. A classical, theoretical problem is to find conditions which implies that the distributions of the filtering process converge towards a unique limit measure. This problem goes back to a paper of D Blackwell for the case when the Markov chain takes its values in a finite set and it goes back to a paper of H Kunita for the case when the state space of the Markov chain is a compact Hausdor space. Recently, due to work by F Kochmann, J Reeds, P Chigansky and R van Handel, a necessary and sucient condition for the convergence of the distributions of the filtering process has been found for the case when the state space is finite. This condition has since been generalised to the case when the state space is denumerable. In this paper we generalise some of the previous results on convergence in distribution to the case when the Markov chain and the observation sequence of a Hidden Markov Model take their values in complete, separable, metric spaces; it has though been necessary to assume that both the transition probability function of the Markov chain and the transition probability function that generates the observation sequence have densities.

Bibliography: leaves 145-150. Electronic publication; Full text available in PDF format; abstract in HTML format. A MAP/PH/1 queue is a queue having a Markov arrival process (MAP), and a single server with phase-type (PH-type) distributed service time. This thesis considers the departure process of these types of queues, using matrix analytic methods, the Jordan canonical form of matrices, non-linear filtering and approximation techniques. Electronic reproduction.[Australia] :Australian Digital Theses Program,2001.