Electronic Communications in Probability

A set-indexed Ornstein-Uhlenbeck process

Paul Balança and Erick Herbin

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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.

Article information

Electron. Commun. Probab., Volume 17 (2012), paper no. 39, 14 pp.

Accepted: 5 September 2012
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G10: Stationary processes
Secondary: 60G15: Gaussian processes 60G60: Random fields 60J25: Continuous-time Markov processes on general state spaces

Ornstein-Uhlenbeck process Markov property multiparameter and set-indexed processes stationarity

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Balança, Paul; Herbin, Erick. A set-indexed Ornstein-Uhlenbeck process. Electron. Commun. Probab. 17 (2012), paper no. 39, 14 pp. doi:10.1214/ECP.v17-1903. https://projecteuclid.org/euclid.ecp/1465263172

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