The Annals of Probability

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

Hardeep Gill

Full-text: Open access


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.

Article information

Ann. Probab., Volume 41, Number 2 (2013), 989-1029.

First available in Project Euclid: 8 March 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J68: Superprocesses
Secondary: 60G57: Random measures

Superprocess interacting measure-valued diffusion Ornstein–Uhlenbeck process center of mass law of large numbers


Gill, Hardeep. A super Ornstein–Uhlenbeck process interacting with its center of mass. Ann. Probab. 41 (2013), no. 2, 989--1029. doi:10.1214/11-AOP741.

Export citation


  • [1] Engländer, J. (2010). The center of mass for spatial branching processes and an application for self-interaction. Electron. J. Probab. 15 1938–1970.
  • [2] Engländer, J. and Turaev, D. (2002). A scaling limit theorem for a class of superdiffusions. Ann. Probab. 30 683–722.
  • [3] Engländer, J. and Winter, A. (2006). Law of large numbers for a class of superdiffusions. Ann. Inst. Henri Poincaré Probab. Stat. 42 171–185.
  • [4] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [5] Garsia, A. M., Rodemich, E. and Rumsey, H. Jr. (1970/1971). A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20 565–578.
  • [6] Gill, H. S. (2009). Superprocesses with spatial interactions in a random medium. Stochastic Process. Appl. 119 3981–4003.
  • [7] Konno, N. and Shiga, T. (1988). Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 201–225.
  • [8] Perkins, E. (1995). On the martingale problem for interactive measure-valued branching diffusions. Mem. Amer. Math. Soc. 115 vi+89.
  • [9] Perkins, E. (2002). Dawson–Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999). Lecture Notes in Math. 1781 125–324. Springer, Berlin.
  • [10] Pinsky, R. G. (1996). Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions. Ann. Probab. 24 237–267.
  • [11] Polyanin, A. D. and Manzhirov, A. V. (2008). Handbook of Integral Equations, 2nd ed. Chapman & Hall/CRC, Boca Raton, FL.
  • [12] Roberts, G. O. and Tweedie, R. L. (2000). Rates of convergence of stochastically monotone and continuous time Markov models. J. Appl. Probab. 37 359–373.
  • [13] Rogers, L. and Williams, D. (1985). Diffusions, Markov Processes and Martingales. Cambridge Univ. Press, Cambridge.
  • [14] Tribe, R. (1992). The behavior of superprocesses near extinction. Ann. Probab. 20 286–311.
  • [15] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.