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 Englander [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 Englander and Winter [Ann. Inst. Henri Poincare 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.