Set-Markov processes.

Abstract

This thesis introduces a type of Markov property, called the 'set-Markov' property, that can be defined for set-indexed processes, and in particular for multiparameter processes. This property is stronger than the 'sharp Markov' property that has been introduced earlier in the literature. For processes indexed by the real line, the set-Markov property coincides with the classical Markov property. An important class of set-Markov processes are ' Q -Markov' processes, where Q is a family of transition probabilities satisfying a Chapman-Kolmogorov type relationship. Two constructions are indicated for a Q -Markov process, as applications of Kolmogorov's extension theorem; the first construction is valid only under a certain geometrical condition. A Q -Markov process can be associated to a 'suitably indexed' collection of classical Markov processes; therefore, 'the generator' of the process is defined as the collection of all the generators of these classical Markov processes. It is proved that under certain conditions, one can construct a Q -Markov process knowing its generator. Processes with independent increments are Q -Markov; they are characterized by 'convolution systems' of distributions. In particular, Levy processes have a double characterization in terms of their characteristic functions and in terms of their generators. Some other examples of Q -Markov processes are considered, including empirical processes. For each of these examples, the generator provides us with a means of constructing the process. 'Adapted sets' and 'optional sets' generalize the classical notions of stopping time and optional time. Using these sets, the strong Markov properties associated to a Q -Markov process, respectively a sharp Markov process, are considered.