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

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.