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.Comment: 13 page