## The Annals of Probability

### Local conditioning in Dawson–Watanabe superprocesses

Olav Kallenberg

#### Abstract

Consider a locally finite Dawson–Watanabe superprocess $\xi=(\xi_{t})$ in $\mathsf{R}^{d}$ with $d\geq2$. Our main results include some recursive formulas for the moment measures of $\xi$, with connections to the uniform Brownian tree, a Brownian snake representation of Palm measures, continuity properties of conditional moment densities, leading by duality to strongly continuous versions of the multivariate Palm distributions, and a local approximation of $\xi_{t}$ by a stationary cluster $\tilde{\eta}$ with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditional distribution of $\xi_{t}$ for a fixed $t>0$, given that $\xi_{t}$ charges the $\varepsilon$-neighborhoods of some points $x_{1},\ldots,x_{n}\in\mathsf{R}^{d}$. In the limit as $\varepsilon\to0$, the restrictions to those sets are conditionally independent and given by the pseudo-random measures $\tilde{\xi}$ or $\tilde{\eta}$, whereas the contribution to the exterior is given by the Palm distribution of $\xi_{t}$ at $x_{1},\ldots,x_{n}$. Our proofs are based on the Cox cluster representations of the historical process and involve some delicate estimates of moment densities.

#### Article information

Source
Ann. Probab., Volume 41, Number 1 (2013), 385-443.

Dates
First available in Project Euclid: 23 January 2013

https://projecteuclid.org/euclid.aop/1358951990

Digital Object Identifier
doi:10.1214/11-AOP702

Mathematical Reviews number (MathSciNet)
MR3059202

Zentralblatt MATH identifier
1270.60058

#### Citation

Kallenberg, Olav. Local conditioning in Dawson–Watanabe superprocesses. Ann. Probab. 41 (2013), no. 1, 385--443. doi:10.1214/11-AOP702. https://projecteuclid.org/euclid.aop/1358951990

#### References

• [1] Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure, 2nd ed. Springer, New York.
• [2] Dawson, D. A. (1993). Measure-valued Markov processes. In École D’Été de Probabilités de Saint-Flour XXI—1991. Lecture Notes in Math. 1541 1–260. Springer, Berlin.
• [3] Dawson, D. A., Iscoe, I. and Perkins, E. A. (1989). Super-Brownian motion: Path properties and hitting probabilities. Probab. Theory Related Fields 83 135–205.
• [4] Dawson, D. A. and Perkins, E. A. (1991). Historical processes. Mem. Amer. Math. Soc. 93 (454) iv+179.
• [5] Dynkin, E. B. (1991). Branching particle systems and superprocesses. Ann. Probab. 19 1157–1194.
• [6] Dynkin, E. B. (1994). An Introduction to Branching Measure-valued Processes. CRM Monograph Series 6. Amer. Math. Soc., Providence, RI.
• [7] Etheridge, A. M. (2000). An Introduction to Superprocesses. University Lecture Series 20. Amer. Math. Soc., Providence, RI.
• [8] Gorostiza, L. G., Roelly-Coppoletta, S. and Wakolbinger, A. (1990). Sur la persistance du processus de Dawson–Watanabe stable. L’interversion de la limite en temps et de la renormalisation. In Séminaire de Probabilités, XXIV, 1988/89. Lecture Notes in Math. 1426 275–281. Springer, Berlin.
• [9] 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.
• [10] Jagers, P. (1973). On Palm probabilities. Z. Wahrsch. Verw. Gebiete 26 17–32.
• [11] Kallenberg, O. (1977). Stability of critical cluster fields. Math. Nachr. 77 7–43.
• [12] Kallenberg, O. (1986). Random Measures, 4th ed. Akademie-Verlag, Berlin.
• [13] Kallenberg, O. (1999). Palm measure duality and conditioning in regenerative sets. Ann. Probab. 27 945–969.
• [14] Kallenberg, O. (2001). Local hitting and conditioning in symmetric interval partitions. Stochastic Process. Appl. 94 241–270.
• [15] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
• [16] Kallenberg, O. (2003). Palm distributions and local approximation of regenerative processes. Probab. Theory Related Fields 125 1–41.
• [17] Kallenberg, O. (2005). Probabilistic Symmetries and Invariance Principles. Springer, New York.
• [18] Kallenberg, O. (2007). Some problems of local hitting, scaling, and conditioning. Acta Appl. Math. 96 271–282.
• [19] Kallenberg, O. (2008). Some local approximations of Dawson–Watanabe superprocesses. Ann. Probab. 36 2176–2214.
• [20] Kallenberg, O. (2009). Some local approximation properties of simple point processes. Probab. Theory Related Fields 143 73–96.
• [21] Kallenberg, O. (2010). Commutativity properties of conditional distributions and Palm measures. Commun. Stoch. Anal. 4 21–34.
• [22] Kallenberg, O. (2011). Iterated Palm conditioning and some Slivnyak-type theorems for Cox and cluster processes. J. Theoret. Probab. 24 875–893.
• [23] Kummer, G. and Matthes, K. (1970). Verallgemeinerung eines Satzes von Sliwnjak. II–III. Rev. Roumaine Math. Pures Appl. 15 845–870, 1631–1642.
• [24] Le Gall, J.-F. (1986). Sur la saucisse de Wiener et les points multiples du mouvement brownien. Ann. Probab. 14 1219–1244.
• [25] Le Gall, J.-F. (1986). Une approche élémentaire des théorèmes de décomposition de Williams. In Séminaire de Probabilités XX. Lecture Notes in Math. 1204, 447–464. Springer, Berlin.
• [26] Le Gall, J.-F. (1991). Brownian excursions, trees and measure-valued branching processes. Ann. Probab. 19 1399–1439.
• [27] Le Gall, J.-F. (1994). A lemma on super-Brownian motion with some applications. In The Dynkin Festschrift. Progress in Probability 34 237–251. Birkhäuser, Boston, MA.
• [28] Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
• [29] Liemant, A., Matthes, K. and Wakolbinger, A. (1988). Equilibrium Distributions of Branching Processes. Mathematical Research 42. Akademie-Verlag, Berlin.
• [30] Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. Wiley, Chichester.
• [31] Mecke, J. (1967). Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. Verw. Gebiete 9 36–58.
• [32] 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.
• [33] Salisbury, T. S. and Verzani, J. (1999). On the conditioned exit measures of super Brownian motion. Probab. Theory Related Fields 115 237–285.
• [34] Salisbury, T. S. and Verzani, J. (2000). Nondegenerate conditionings of the exit measures of super Brownian motion. Stochastic Process. Appl. 87 25–52.
• [35] Slivnyak, I. M. (1962). Some properties of stationary flows of homogeneous random events. Theory Probab. Appl. 7 336–341; 9 168.
• [36] Williams, D. (1974). Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. London Math. Soc. (3) 28 738–768.
• [37] Zähle, U. (1988). The fractal character of localizable measure-valued processes. II. Localizable processes and backward trees. Math. Nachr. 137 35–48.