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Journal Articles Electronic Communications in Probability Year : 2012

## A set-indexed Ornstein-Uhlenbeck process

Paul Balança
Erick Herbin

#### Abstract

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.

#### Domains

Mathematics [math] Probability [math.PR]

### Dates and versions

hal-00734421 , version 1 (21-09-2012)

### Identifiers

• HAL Id : hal-00734421 , version 1
• ARXIV :
• DOI :

### Cite

Paul Balança, Erick Herbin. A set-indexed Ornstein-Uhlenbeck process. Electronic Communications in Probability, 2012, 17 (39), pp.1-14. ⟨10.1214/ECP.v17-1903⟩. ⟨hal-00734421⟩

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