Section: New Results
An increment type set-indexed Markov property
Participant : Paul Balança.
 investigates a new approach for the definition of a set-indexed Markov property, named -Markov. The study is based on Merzbach and Ivanoff's set-indexed formalism, i.e. denotes a set-indexed collection and the family of increments , where and (finite unions of sets from ). Moreover, for any , , is defined as the following subset of :
and corresponds to the semilattice . The notation refers to a random vector . Similarly, is used to denote a vector of variables .
Then, an -valued set-indexed process is said to be -Markov with respect to a filtration if it is adapted to and if it satisfies
for all and any bounded measurable function . The sigma-algebra is usually called the strong history of and is defined as .
The -Markov approach has several advantages compared to existing set-indexed Markov literature (mainly -Markov described in  ). It appears to be a natural extension of the classic one-parameter Markov property. In particular, the concept of transition system can easily extended to our formalism: for any -Markov process , one can defined as
A -transition system happens to satisfy a set-indexed Chapman-Kolmogorov equation,
and is a bounded measurable function.
Similarly to the classic Markovian theory, is is proved in  that the initial distribution and characterize entirely the law of a -Markov process, and that conversely, for any initial law and any -transition system, a corresponding canonical set-indexed -Markov process can be constructed. -Markov processes enjoy several other properties such as
The class of set-indexed Lévy processes defined and studied in  offers examples of -Markov processes whose transition probabilities correspond to
where is a measure on and the infinitely divisible probability measure that characterizes the Lévy process. We note that the transition system related the -Markov property has a different form, even if it is related.
Another non-trivial example of -Markov process is the set-indexed Ornstein-Uhlenbeck process that has been introduced and studied in  . It is a Gaussian Markovian process whose transition densities are given by
where and are positive parameters, is a measure on and
In the particular case of multiparameter processes, corresponding to the indexing collection , the -Markov formalism is related to several existing works. It generalizes the two-parameter -Markov property introduced in  and also embraces the multiparameter Markov property investigated recently in  . Finally, under some Feller assumption on the transition system, a multiparameter -Markov process is proved to admit a modification with right-continuous sample paths.