## The Annals of Probability

### Invariant measures of critical spatial branching processes in high dimensions

#### Abstract

We consider two critical spatial branching processes on $\mathbb{R}^d$: critical branching Brownian motion, and the critical Dawson-Watanabe process. A basic feature of these processes is that their ergodic behavior is highly dimension dependent. It is known that in low dimensions, $d \leq 2$, the only invariant measure is $\delta_0$ , the unit point mass on the empty state. In high dimensions, $d \geq 3$, there is a family ${\nu_{\theta}, \theta \epsilon [0, \infty)}$ of extremal invariant measures; the measures ${\nu_{\theta}$ are translation invariant and indexed by spatial intensity. We prove here, for $d \geq 3$, that all invariant measures are convex combinations of these measures.

#### Article information

Source
Ann. Probab., Volume 25, Number 1 (1997), 56-70.

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1024404278

Digital Object Identifier
doi:10.1214/aop/1024404278

Mathematical Reviews number (MathSciNet)
MR1428499

Zentralblatt MATH identifier
0882.60091

#### Citation

Bramson, Maury; Cox, J. T.; Greven, Andreas. Invariant measures of critical spatial branching processes in high dimensions. Ann. Probab. 25 (1997), no. 1, 56--70. doi:10.1214/aop/1024404278. https://projecteuclid.org/euclid.aop/1024404278

#### References

• 1 AIZENMAN, M. 1980. Translation invariance and instability of phase coexistence in the two-dimensional Ising system. Comm. Math. Phys. 73 83 94.
• 2 BRAMSON, M., COX, J. T. and GREVEN, A. 1993. Ergodicity of critical spatial branching processes in low dimensions. Ann. Probab. 21 1946 1957.
• 3 COX, J. T. 1994. On the ergodic theory of critical branching Markov chains. Stochastic Process. Appl. 50 1 20.
• 4 DAWSON, A. 1977. The critical measure diffusion process.Wahrsch. Verw. Gebiete 40 125 145.
• 5 DAWSON, D. A. 1993. Measure-valued Markov processes. Ecole d'ete de Probabilites de ´ ´ ´ Saint-Flour, Lecture Notes in Math. 1541 1 260. Springer, New York.
• 6 DAWSON, D. A. and IVANOFF, B. G. 1978. Branching diffusions and random measures. Advances in Probability: Branching Processes 61 109.
• 7 DURRETT, R. 1979. An infinite particle system with additive interactions. Adv. in Appl. Probab. 11 355 383.
• 8 DYNKIN, E. B. 1989. Three classes of infinite dimensional diffusions. J. Funct. Anal. 86 75 110.
• 9 ETHIER, S. and KURTZ, T. 1986. Markov Processes: Characterization and Convergence. Wiley, New York.
• 10 FLEISCHMAN, J. 1978. Limiting distributions for branching random fields. Trans. Amer. Math. Soc. 239 353 389.
• 11 GOROSTIZA, L. G. and WAKOLBINGER, A. 1994. Long time behavior of critical branching particle systems and applications. CRM Proceedings and Lecture Notes 5 119 138.
• 12 ISCOE, I. 1986. A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Related Fields 71 85 116.
• 13 KERSTAN, J., MATTHES, K. and MECKE, J. 1978. Infinitely Divisible Point Processes, Wiley, New York.
• 14 KRENGEL, U. 1985. Ergodic Theorems. de Gruyter, Berlin.
• 15 LIGGETT, T. M. 1985. Interacting Particle Systems. Springer, New York.
• 16 NGUYEN, X. X. and ZESSIN, H. 1979. Ergodic theorems for spatial processes.Wahrsch. Verw. Gebeite 48 133 158.
• 17 PERKINS, E. A. 1988. A space-time property of a class of measure-valued branching diffusions. Trans. Amer. Math. Soc. 305 743 794.
• 18 PROTTER, M. H. and WEINBERGER, H. F. 1967. Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs, NJ.
• 21 STROOCK, D. W. and VARADHAN, S. R. S. 1979. Multidimensional Diffusion Processes. Springer, New York.
• MADISON, WISCONSIN 53706 SYRACUSE, NEW YORK 13244 E-MAIL: bramson@math.wisc.edu E-MAIL: jtcox@gumby.syr.edu