Day on Lévy processes

Consider a spectrally one-sided Lévy process and reflect it at its infimum. First we consider the Laplace transform of the first exit time of this reflected process from a finite interval [0,a) and show it can be expressed in terms of (adjoint) scale funtions. Next we turn our attention to the ergodic properties of the transition probabilities of this reflected process.

Using the Wiener-Hopf factorisation it is shown that it is possible to bound the path of an arbitrary Lévy process above and below by the paths of two random walks. These walks have the same step distribution, but different random starting points. In principle, this allows one to deduce Lévy process versions of many known results about the large-time behaviour of random walks. This is illustrated by some results about Lévy processes which converge to infinity in probability.

Marc will talk about Kennedy martingales and their use in establishing fluctuation identities for Browinan motion.

Two and one sided exit problems for spectrally negative Lévy processes have been the object of several studies over the last 40 years. Significant contributions have come from Zolotarev (1964), Takacs (1967), Emery (1973), Bingham (1975), Rogers (1990) and Bertoin (1996a, 1996b, 1997). The principal tools of analysis of these authors are the Wiener-Hopf factorization and It\^{o}'s excursion theory. The aim of this talk is to give a reasonably self contained approach to some elementary fluctuation theory which avoids the use of specialist theorems. Specifically we avoid the use of the Wiener-Hopf factorization and It\^{o}'s excursion theory and rely mainly on martingale arguments together with the Strong Markov property. None of the results we present are new but for the most part, the proofs approach the results from a new angle bearing a similar flavour to the Kennedy martingales given in the previous talk.

A composition structure is a sequence of consistent random compositions (ordered partitions) of integers or, what is essentially the same, an exchangeable ordered partition of an infinitely countable set. We introduce a class of {\it regenerative} composition structures characterised by the first-part deletion property, which resembles Kingman's random-part deletion (that lead to the Ewens partition structure). Regenerative compositions constitute a rich family appearing naturally by a `paintbox' construction involving multiplicatively regenerative random sets, which in turn can be seen as a result of generalised stick-breaking. These compositions are described by a product sampling formula with constituents easily expressible via Lévy parameters of underlying subordinator. Interesting examples include compositions associated with the Ewens-Pitman two-parameter family of partition structures, and some new sampling formulas related to beta distributions. Recursive character of compositions makes possible application of divide-and-conquer techniques to the analysis of most important functionals like the number of parts.

Homogeneous fragmentations describe the evolution of a mass that breaks down into pieces as time passes. They can be thought of as continuous time analogs of branching random walks. Using Kingman's representation of exchangeable partitions of $\N$, we adapt to fragmentations the method of probability tilting of Lyons, Pemantle and Peres. Subordinators play a major role in this approach.