The set-indexed Lévy process: Stationarity, Markov and sample paths properties

We present a satisfactory definition of the important class of Lévy processes indexed by a general collection of sets. We use a new definition for increment stationarity of set-indexed processes to obtain different characterizations of this class. As an example, the set-indexed compound Poisson process is introduced. The set-indexed Lévy process is characterized by infinitely divisible laws and a Lévy–Khintchine representation. Moreover, the following concepts are discussed: projections on flows, Markov properties, and pointwise continuity. Finally the study of sample paths leads to a Lévy–Itô decomposition. As a corollary, the semi-martingale property is proved.