The Annals of Probability

A super Ornstein–Uhlenbeck process interacting with its center of mass

Hardeep Gill

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Abstract

We construct a supercritical interacting measure-valued diffusion with representative particles that are attracted to, or repelled from, the center of mass. Using the historical stochastic calculus of Perkins, we modify a super Ornstein–Uhlenbeck process with attraction to its origin, and prove continuum analogues of results of Engländer [Electron. J. Probab. 15 (2010) 1938–1970] for binary branching Brownian motion.

It is shown, on the survival set, that in the attractive case the mass normalized interacting measure-valued process converges almost surely to the stationary distribution of the Ornstein–Uhlenbeck process, centered at the limiting value of its center of mass. In the same setting, it is proven that the normalized super Ornstein–Uhlenbeck process converges a.s. to a Gaussian random variable, which strengthens a theorem of Engländer and Winter [Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 171–185] in this particular case. In the repelling setting, we show that the center of mass converges a.s., provided the repulsion is not too strong and then give a conjecture. This contrasts with the center of mass of an ordinary super Ornstein–Uhlenbeck process with repulsion, which is shown to diverge a.s.

A version of a result of Tribe [Ann. Probab. 20 (1992) 286–311] is proven on the extinction set; that is, as it approaches the extinction time, the normalized process in both the attractive and repelling cases converges to a random point a.s.

Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 989-1029.

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1362750948

Digital Object Identifier
doi:10.1214/11-AOP741

Mathematical Reviews number (MathSciNet)
MR3077532

Zentralblatt MATH identifier
1279.60110

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

Keywords
Superprocess interacting measure-valued diffusion Ornstein–Uhlenbeck process center of mass law of large numbers

Citation

Gill, Hardeep. A super Ornstein–Uhlenbeck process interacting with its center of mass. Ann. Probab. 41 (2013), no. 2, 989--1029. doi:10.1214/11-AOP741. https://projecteuclid.org/euclid.aop/1362750948


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References

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