Balança, Paul; Herbin, Erick A set-indexed Ornstein-Uhlenbeck process. (English) Zbl 1266.60066 Electron. Commun. Probab. 17, Paper No. 39, 14 p. (2012). Summary: 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 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. Cited in 2 Documents MSC: 60G10 Stationary stochastic processes 60G15 Gaussian processes 60G60 Random fields 60J25 Continuous-time Markov processes on general state spaces Keywords:Ornstein-Uhlenbeck process; Markov property; multiparameter and set-indexed processes; stationarity × Cite Format Result Cite Review PDF Full Text: DOI arXiv