On etudie la valeur principale de Cauchy des temps locaux du mouvement brownien lineaire. En particulier, on donne sa loi pour diverses valeurs du temps; les calculs utilisent la theorie des excursions du mouvement brownien et menent a des resultats simples, relies a une formule de P. Levy

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Valeurs principales associées aux temps locaux browniens

An explicit solution to the Skorokhod embedding problem for functionals of excursions of Markov processes

We develop an explicit non-randomized solution to the Skorokhod embedding problem in an abstract setup of signed functionals of Markovian excursions. Our setting allows to solve the Skorokhod embedding problem, in particular, for diffusions and their (signed, scaled) age processes, for Azema's martingale, for spectrally one-sided Levy processes and their reflected versions, for Bessel processes of dimension smaller than $2$, and for their age processes, as well as for the age process of excursions of Cox-Ingersoll-Ross processes. This work is a continuation and an important generalization of Obloj and Yor (SPA 110) [35]. Our methodology, following [35], is based on excursion theory and the solution to the Skorokhod embedding problem is described in terms of the Ito measure of the functional. We also derive an embedding for positive functionals and we correct a mistake in the formula in [35] for measures with atoms.

An explicit Skorokhod embedding for functionals of Markovian excursions

"Sur la théorie des excursions pour des processus de Lévy symétriques stables d'indice 2)1,2), et quelques applications."

Expressions for the generalized covariances of multi-dimensional Brownian excursion local times are derived from corresponding densities transforms. Typical applications are moments of the cost of structures such as M /G /1 queue, Random trees, Markov stack or priority queue in Knuth’s model. Brownian excursion area and a result of Biane and Yor are also revisited.

Generalized Covariances of Multi-dimensional Brownian Excursion Local Times

We study the following one-dimensional model of annihilating particles. Beginning with all sites of $\mathbb{Z}$ uncolored, a blue particle performs simple random walk from $0$ until it reaches a nonzero red or uncolored site, and turns that site blue; then, a red particle performs simple random walk from $0$ until it reaches a nonzero blue or uncolored site, and turns that site red. We prove that after $n$ blue and $n$ red particles alternately perform such walks, the total number of colored sites is of order $n^{1/4}$. The resulting random color configuration, after rescaling by $n^{1/4}$ and taking $n\to \infty$, has an explicit description in terms of alternating extrema of Brownian motion (the global maximum on a certain interval, the global minimum attained after that maximum, etc.).

Valeurs principales associées aux temps locaux browniens

Citations

An explicit solution to the Skorokhod embedding problem for functionals of excursions of Markov processes

An explicit Skorokhod embedding for functionals of Markovian excursions

"Sur la théorie des excursions pour des processus de Lévy symétriques stables d'indice 2)1,2), et quelques applications."

Generalized Covariances of Multi-dimensional Brownian Excursion Local Times

Formation of large-scale random structure by competitive erosion

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