A set-indexed Ornstein-Uhlenbeck process

Details A set-indexed Ornstein-Uhlenbeck process A set-indexed Ornstein-Uhlenbeck process

2012-03-25

arxiv.org/abs/1203.5524v1

Mathematics

Probability

13 pages

Scientific paper

The purpose of this article is a set-indexed extension of the well-known Ornstein-Uhlenbeck process. The first part is devoted to a stationary definition of the random field and ends up with the proof of a complete characterization by its $L^2$-continuity, stationarity and set-indexed Markov properties. This specific Markov transition system allows to define a general \emph{set-indexed Ornstein-Uhlenbeck (SIOU) process} with any initial probability measure. Finally, in the multiparameter case, the SIOU process is proved to admit a natural integral representation.

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