9 Rue Charles Fourier, 91000 \u00c9vry-Courcouronnes<\/a><\/p>\n\n\n\nMembres du jury :<\/strong><\/p>\n\n\n\nM. Sandro FRANCESCHI<\/strong>, Ma\u00eetre de conf\u00e9rences, Institut Polytechnique de Paris T\u00e9l\u00e9com SudParis, FRANCE – Directeur de th\u00e8se
Mme H\u00e9l\u00e8ne GU\u00c9RIN<\/strong>, Professeure, Universit\u00e9 du Qu\u00e9bec \u00e0 Montr\u00e9al, CANADA – Rapporteur
M. Julien ROQUES<\/strong>, Professeur, Lyon 1 Universit\u00e9, FRANCE – Rapporteur
M. Alin BOSTAN<\/strong>, Directeur de recherche, Universit\u00e9 de Paris-Saclay (INRIA), FRANCE – Examinateur
Mme Manon DEFOSSEUX<\/strong>, Ma\u00eetresse de conf\u00e9rences, Universit\u00e9 Paris Cit\u00e9, FRANCE – Examinateur
Mme Eva L\u00d6CHERBACH<\/strong>, Professeure, Universit\u00e9 Paris 1 Panth\u00e9on-Sorbonne, FRANCE – Examinateur
Mme Sara MAZZONETTO<\/strong>, Ma\u00eetresse de conf\u00e9rences, Universit\u00e9 de Lorraine, FRANCE – Examinateur<\/p>\n\n\n\nInvit\u00e9 :<\/strong><\/p>\n\n\n\nMme Marie ALBENQUE,<\/strong> Directrice de recherche, Universit\u00e9 Paris Cit\u00e9, FRANCE – Codirectrice de th\u00e8se<\/p>\n\n\n\n\u00ab Invariants de Tutte, probl\u00e8mes fronti\u00e8re et mouvement brownien r\u00e9fl\u00e9chi \u00bb<\/h2>\n\n\n\npr\u00e9sent\u00e9 par Monsieur Jules FLIN<\/h2>\n\n\n\n
R\u00e9sum\u00e9 :<\/strong><\/p>\n\n\n\nLe mouvement brownien r\u00e9fl\u00e9chi dans un quadrant, et plus g\u00e9n\u00e9ralement dans des c\u00f4nes poly\u00e9driques, a \u00e9t\u00e9 introduit par Harrison, Reiman, Varadhan et Williams dans les ann\u00e9es 1980 comme limite d’\u00e9chelle en th\u00e9orie des files d’attente. Depuis lors, l’\u00e9tude th\u00e9orique de ce processus suscite une grande attention au sein de la communaut\u00e9 probabiliste. Plus r\u00e9cemment, une m\u00e9thode issue de la combinatoire, initialement d\u00e9velopp\u00e9e par Tutte et bas\u00e9e sur la notion d’invariant, a \u00e9t\u00e9 adapt\u00e9e \u00e0 l’\u00e9tude de ces diffusions r\u00e9fl\u00e9chies. Cette approche s’av\u00e8re particuli\u00e8rement puissante pour \u00e9tudier les \u00e9quations fonctionnelles satisfaites par les transform\u00e9es de Laplace de certaines fonctions associ\u00e9es au processus. La m\u00e9thode des invariants de Tutte permet en effet d’en obtenir des solutions explicites dans un grand nombre de cas et, en la combinant \u00e0 la th\u00e9orie de Galois des \u00e9quations aux diff\u00e9rences, de d\u00e9terminer la nature diff\u00e9rentielle et alg\u00e9brique de ces transform\u00e9es, autrement dit, la classe d’\u00e9quations diff\u00e9rentielles dont elles sont solutions. Dans un contexte o\u00f9 le processus peut \u00eatre absorb\u00e9 \u00e0 l’origine en temps fini, nous montrons au Chapitre 2 comment cette approche permet de calculer explicitement la probabilit\u00e9 d’un tel \u00e9v\u00e9nement. Le Chapitre 3 est d\u00e9di\u00e9 \u00e0 la d\u00e9termination de la mesure invariante du processus, lorsque sa matrice de covariance est d\u00e9g\u00e9n\u00e9r\u00e9e. Enfin, dans le Chapitre 4, nous calculons la mesure invariante d’un mouvement brownien r\u00e9fl\u00e9chi dans le demi-plan sup\u00e9rieur, soumis \u00e0 des r\u00e9flexions constantes de part et d’autre de l’origine. Pour ce faire, nous nous appuyons sur la th\u00e9orie des probl\u00e8mes fronti\u00e8re. Cette approche analytique nous permet \u00e9galement d’\u00e9tudier pr\u00e9cis\u00e9ment les asymptotiques de sa densit\u00e9.
Abstract :<\/strong><\/p>\n\n\n\nReflected Brownian motion in a quadrant, and more generally in polyhedral cones, was introduced by Harrison, Reiman, Varadhan, and Williams in the 1980s as a scaling limit in queueing theory. Since then, the theoretical study of this process has attracted significant attention within the probability community. More recently, a combinatorial method originally developed by Tutte and based on the notion of invariants was adapted to the study of these reflected diffusions. This approach proves particularly powerful for studying the functional equations satisfied by the Laplace transforms of certain functions associated with the process. Indeed, Tutte’s invariant method yields explicit solutions to them in a large number of cases and, when combined with the Galois theory of difference equations, enables the determination of the differential and algebraic nature of these transforms, that is, the class of differential equations of which they are solutions. In a context where the process can be absorbed at the origin in finite time, we show in Chapter 2 how this approach allows for the explicit computation of the probability of such an event. Chapter3 is dedicated to determining the invariant measure of the process when its covariance matrix is degenerate. Finally, in Chapter 4, we compute the invariant measure of a reflected Brownian motion in the upper half-plane, subject to constant reflections on both sides of the origin. To do so, we rely on the theory of boundary value problems. This analytical approach also enables us to precisely study the asymptotics of its density.<\/p>\n","protected":false},"excerpt":{"rendered":"
L’\u00c9cole doctorale : Math\u00e9matiques Hadamard et le Laboratoire de recherche SAMOVAR – Services r\u00e9partis, Architectures, Mod\u00e9lisation, Validation, Administration des R\u00e9seaux pr\u00e9sentent l\u2019AVIS DE SOUTENANCE de Monsieur Jules FLIN Autoris\u00e9 \u00e0 pr\u00e9senter ses travaux en vue de l\u2019obtention du Doctorat de l’Institut Polytechnique de Paris, pr\u00e9par\u00e9 \u00e0 l’Institut Polytechnique de Paris T\u00e9l\u00e9com SudParis en : Math\u00e9matiques 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