Jean Bertoin

On continuity properties of the law of integrals of levy processes

Let (ξ, η) be a bivariate Levy process such that the integral \(\int_0^\infty {e^{ - \xi _{t - } } d\eta _t }\)converges almost surely. We characterise, in terms of their Levy measures, those Levy processes for which (the distribution of) this integral has atoms. We then turn attention to almost surely convergent integrals of the form I := ∫ 0 ∞ g(ξ t ) dt, where g is a deterministic function. We give sufficient conditions ensuring that I has no atoms, and under further conditions derive that I has a Lebesgue density. The results are also extended to certain integrals of the form ∫ 0 ∞ g(ξ t ) dY t , where Y is an almost surely strictly increasing stochastic process, independent of ξ.

Spitzer's condition for random walks and Lévy Processes

Spitzers condition holds for a random walk S if the probabilities ρn=P{Sn> 0} converge in Cesaro mean to ρ, and for a Levy process X at ∞ (at 0, respectively) if t1 ∫0t ρ(s)ds→ ρ as t→ ∞(0), where ρ(s)=P{Xs >0}. It has been shown in Doney [4] that if 0 < ρ < 1 then this happens for a random walk if and only if ρn converges to ρ. We show here that this result extends to the cases ρ = 0 and ρ = 1, and also that Spitzers condition holds for a Levy process at ∞(0) if and only if ρ(t) → ρ as t → ∞(0).

Fires on trees

We consider random dynamics on the edges of a uniform Cayley tree with

Some applications of duality for Lévy processes in a half-line

n

Reflecting a langevin process at an absorbing boundary

vertices, in which edges are either inflammable, fireproof, or burt. Every inflammable edge is replaced by a fireproof edge at unit rate, while fires start at smaller rate

Une extension d'une inégalité de burkholder, davis, gundy pour les processus à α-variation bornée et applications

n^{-\alpha}

A Ratio Ergodic Theorem for Brownian Additive Functionals with Infinite Mean

on each inflammable edge, then propagate through the neighboring inflammable edges and are only stopped at fireproof edges. A vertex is called fireproof when all its adjacent edges are fireproof. We show that as

Some Aspects of Random Fragmentations in Continuous Times

n\to \infty

Stochastic flows associated to coalescent processes. III. Limit theorems

, the density of fireproof vertices converges to

On the First Exit Time of a Completely Asymmetric Stable Process from a Finite Interval

1