  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.
