## Electronic Journal of Probability

### Probability measure-valued polynomial diffusions

#### Abstract

We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming–Viot process is a particular example. The defining property of finite dimensional polynomial processes considered in [8, 21] is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 30, 32 pp.

Dates
Accepted: 2 March 2019
First available in Project Euclid: 26 March 2019

https://projecteuclid.org/euclid.ejp/1553565781

Digital Object Identifier
doi:10.1214/19-EJP290

Mathematical Reviews number (MathSciNet)
MR3933209

Zentralblatt MATH identifier
07055668

Subjects
Primary: 60J68: Superprocesses 60G57: Random measures

#### Citation

Cuchiero, Christa; Larsson, Martin; Svaluto-Ferro, Sara. Probability measure-valued polynomial diffusions. Electron. J. Probab. 24 (2019), paper no. 30, 32 pp. doi:10.1214/19-EJP290. https://projecteuclid.org/euclid.ejp/1553565781

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