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

TL;DR

This paper constructs and analyzes a supercritical measure-valued process interacting with its center of mass, revealing convergence behaviors in attractive and repelling cases, and extending classical results in stochastic processes.

Contribution

It introduces a new interacting super Ornstein-Uhlenbeck process with attraction or repulsion to the center of mass, providing continuum analogues of known binary branching Brownian motion results.

Findings

In the attractive case, the normalized process converges to the stationary distribution of OU.

The process converges almost surely to a Gaussian in the attractive case.

Center of mass converges under certain conditions in the repelling case.

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.