Random coverings and self-similar Markov processes

Abstract : This thesis is composed of two parts. The first deals with the construction of a random set which has the property of regeneration. Precisely, we construct random intervals from the partial records of a Poisson point process; these are used to partially cover $\mathbb(R)^+.$ The purpose of this work is to study the random set $\Rs$ that is left uncovered. We give integral tests to decide whether the random set $\Rs$ has a positive Lebesgue measure, has isolated points or if it is bounded. We show that $\Rs$ is, indeed, a regenerative set and characterize its law via the potential measure of the subordinator associated to $\Rs$. We obtain formulas to estimate some fractal dimensions of $\Rs.$ The second part consists of some contributions to the theory of positive self--similar Markov processes. To obtain the results of this part, we use Lamperti's transformation which establishes a bijection between this class of processes and real--valued Lévy processes. Firstly, we are interested in the behavior at infinity of increasing self--similar Markov processes. In this vein, under some hypotheses, we find a deterministic function $f$ such that the liminf, as $t$ goes to infinity, of the quotient $X_t/f(t)$ is finite and different from 0 with probability $1.$ We obtain an analogous result which determines the behavior near of 0 of the process $X$ started from 0. Secondly, we study the different ways to construct a positive self--similar Markov process $\widetilde(X)$ for which 0 is a regular and recurrent point. To this end, we give some conditions that enable us to ensure that a such process exists and to determine its resolvent. Next, we make a systematic study of the Itô excursion measure $\exc$ of the process $\widetilde(X)$. In particular, we give a description of $\exc$ similar to that of Imhof for Itô's excursion measure of Brownian motion; we determine the law under $\exc$ of the normalized excursion and the image under time reversal of $\exc$. Furthermore, we construct and describe a process which is in weak duality with the process $\widetilde(X).$ We obtain some estimations of tail probabilities of the law of an exponential functional of a Lévy process.
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Mathematics. Université Pierre et Marie Curie - Paris VI, 2004. English


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Submitted on : Tuesday, November 9, 2004 - 12:51:54 PM
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Victor RIVERO MERCADO. Random coverings and self-similar Markov processes. Mathematics. Université Pierre et Marie Curie - Paris VI, 2004. English. <tel-00007346>

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