Subordinators, Lévy processes with no negative jumps, and branching processes

The purpose of this course is to present some simple relations connecting subordinators, Lévy processes with no negative jumps, and continuous state branching processes. To start with, we develop the main ingredients on subordinators (the LévyKhintchine formula, the Lévy-Itô decomposition, the law of the iterated logarithm, the renewal theory for the range, and the link with local times of Markov processes). Then we consider Lévy processes with no negative jumps, first in the simple case given by a subordinator with negative drift, and then in the case with unbounded variation. The main formulas of fluctuation theory are presented in this setting, including those related to the so-called two-sided exit problem. Last, we turn our attention to continuous state branching processes. We first discuss the construction by Lamperti based on a simple timesubstitution of a Lévy process with no negative jumps. Then we dwell on the connection with Bochner’s subordination for subordinators and its application to the genealogy of continuous state branching processes.

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