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We construct a supercritical interacting measure-valued diffusion withrepresentative particles that are attracted to, or repelled from, the center ofmass. Using the historical stochastic calculus of Perkins, we modify a superOrnstein-Uhlenbeck process with attraction to its origin, and prove continuumanalogues of results of Englander [Electron. J. Probab. 15 (2010) 1938-1970]for binary branching Brownian motion. It is shown, on the survival set, that inthe attractive case the mass normalized interacting measure-valued processconverges almost surely to the stationary distribution of theOrnstein-Uhlenbeck process, centered at the limiting value of its center ofmass. In the same setting, it is proven that the normalized superOrnstein-Uhlenbeck process converges a.s. to a Gaussian random variable, whichstrengthens a theorem of Englander and Winter [Ann. Inst. Henri PoincareProbab. Stat. 42 (2006) 171-185] in this particular case. In the repellingsetting, we show that the center of mass converges a.s., provided the repulsionis not too strong and then give a conjecture. This contrasts with the center ofmass of an ordinary super Ornstein-Uhlenbeck process with repulsion, which isshown 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 theextinction time, the normalized process in both the attractive and repellingcases converges to a random point a.s.

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