Etude des temps locaux browniens

Abstract

Consider a standard Markov process and a recurrent state x. We study the local time and the set of times where the process passes through x. We generalize works on Brownian motion. With the summation formula of Euler Maclaurin and harmonic analysis, a better control of the number of dyadic boxes entered by the subordinator before some fixed time is possible. Our construction of the local time is new and should be compared to the works of Kingman, Fristedt and Taylor Griego etc.. Our result is intrinsic and is not, for example, a consequence of excursion statistics. Our functional is well suited for Monte Carlo methods. On the other hand, we also contribute to the theory of random multifunctions (and detailed analysis of the random fractal of the level times) by studying the Lebesgue measure of Minkowski sums of random sets. A new index is introduced. This shows the existence of an interesting cut-off phenomenon: the Lebesgue measure of adequate Minkowski linear combinations of k independent copies, k ≥ 1, of the level set is zero then suddenly jumps to a positive value when k crosses the index. It follows for example that for Brownian motion, for r fixed, we can’t find two zeros distant by r. Our technique is based on an analysis of Hausdorff measures. Our results can also be independently developed and used in other branches of probability than the theory of Markov processes itself: in the theory of renewal and regenerative sets for example, or in the synthesis of Itô.