The Annals of Probability

Clustering and invariant measures for spatial branching models with infinite variance

Achim Klenke

Full-text: Open access


We consider two spatial branching models on $\mathbb{R}^d$: branching Brownian motion with a branching law in the domain of normal attraction of a $1 + \beta$ stable law, $0 < \beta \leq 1$, and the corresponding high density limit measure valued diffusion. The longtime behavior of both models depends highly on $\beta$ and $d$. We show that for $d \leq \frac{2}{\beta}$ the only invariant measure is $\delta_0$, the unit mass on the empty configuration. Furthermore, we give a precise condition for convergence toward $\delta_0$. For $d > \frac{2}{\beta}$ it is known that there exists a family $(\nu_{\theta}, \theta \epsilon [0, \infty))$ of nontrivial invariant measures. We show that every invariant measure is a convex combination of the $\nu_{\theta}$. Both results have been known before only under an additional finite mean assumption. For the critical dimension $d = \frac{2}{\beta}$ we show that both models display the phenomenon of diffusive clustering. This means that clusters grow spatially on a random scale. We give a precise description of the clusters via multiple scale analysis. Our methods rely mainly on studying sub- and supersolutions of the reaction diffusion equation $\frac{\partial u}{\partial t} - 1/2 \Delta u + u^{1 + \beta} = 0$.

Article information

Ann. Probab., Volume 26, Number 3 (1998), 1057-1087.

First available in Project Euclid: 31 May 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G57: Random measures

Branching Brownian motion superprocess invariant measures diffusive clustering stable laws reaction-diffusion equation


Klenke, Achim. Clustering and invariant measures for spatial branching models with infinite variance. Ann. Probab. 26 (1998), no. 3, 1057--1087. doi:10.1214/aop/1022855745.

Export citation


  • Bramson, M., Cox, J. T. and Greven, A. (1993). Ergodicity of critical spatial branching processes in low dimensions. Ann. Probab. 21 1946-1957.
  • Bramson, M., Cox, J. T. and Greven, A. (1997). Invariant measures in critical spatial branching processes in high dimensions. Ann. Probab. 25 56-70.
  • Brezis, H. and Friedman, A. (1983). Nonlinear parabolic equations involving measures as initial conditions. J. Math. Pures Appl. 62 73-97.
  • Brezis, H., Peletier, L. A. and Terman, D. (1986). A very singular solution of the heat equation with absorption. Arch. Rational Mech. Anal. 95 185-209.
  • Cox, J. T. and Griffeath, D. (1986). Diffusive clustering in the two-dimensional voter model. Ann. Probab. 14 347-370.
  • Dawson, D. (1993). Measure-valued Markov processes. In Ecole d'Et´e de Probabilit´es de St. Flour XXI. Lecture Notes in Math. 1541 1-260. Springer, Berlin.
  • Fleischmann, K. (1988). Critical behavior of some measure-valued processes. Math. Nachr. 135 131-147.
  • Gorostiza, L. G., Roelly-Coppoletta, S. and Wakolbinger, A. (1990). Sur la persistence du processus de Dawson Watanabe stable (l'interversion de la limite en temps et de la renormalisation). Seminaire de Probabilit´es XXIV. Lecture Notes in Math. 1426 275- 281. Springer, Berlin.
  • Gorostiza, L. G., Roelly, S. and Wakolbinger, A. (1992). Persistence of critical multitype particle systems and measure branching processes. Probab. Theory Related Fields 92 313- 335.
  • Gorostiza, L. G. and Wakolbinger, A. (1991). Persistence criteria for a class of critical branching particle systems in continuous time. Ann. Probab. 19 266-288.
  • Gorostiza, L. G. and Wakolbinger, A. (1992). Convergence to equilibrium of critical branching particle systems and superprocesses, and related nonlinear partial differential equations. Acta Appl. Math. 27 269-291.
  • Kallenberg, O. (1983). Random Measures. Academic Press, New York.
  • Klenke, A. (1996). Different clustering regimes in systems of hierarchically interacting diffusions. Ann. Probab. 24 660-697.
  • Klenke, A. (1997). Multiple scale analysis of clusters in spatial branching models. Ann. Probab. 25 1670-1711.
  • Lee, T.-Y. (1991). Conditional limit distributions of critical branching Brownian motions. Ann. Probab. 19 289-311.
  • Le Gall, J.-F. and Le Jan, Y. (1998). Branching processes in L´evy processes: The exploration process. Ann. Probab. 26 213-252.
  • Protter, M. H. and Weinberger, H. F. (1967). Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs, NJ.