Some local approximations of Dawson–Watanabe superprocesses
Olav Kallenberg
Source: Ann. Probab. Volume 36, Number 6
(2008), 2176-2214.
Abstract
Let ξ be a Dawson–Watanabe superprocess in ℝd such that ξt is a.s. locally finite for every t≥0. Then for d≥2 and fixed t>0, the singular random measure ξt can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the ɛ-neighborhoods of supp ξt. When d≥3, the local distributions of ξt near a hitting point can be approximated in total variation by those of a stationary and self-similar pseudo-random measure ξ̃. By contrast, the corresponding distributions for d=2 are locally invariant. Further results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities. Our main proofs are based on a detailed analysis of the historical structure of ξ.
First Page:
Show
Hide
Keywords: Measure-valued branching diffusions; super-Brownian motion; historical clusters; local distributions; neighborhood measures; hitting probabilities; Palm distributions; self-similarity; moment densities; local extinction
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1229696600
Digital Object Identifier: doi:10.1214/07-AOP386
Mathematical Reviews number (MathSciNet): MR2478680
Zentralblatt MATH identifier: 1167.60010
References
[1] Bramson, M., Cox, J. T. and Greven, A. (1993). Ergodicity of critical spatial branching processes in low dimensions. Ann. Probab. 21 1946–1957.
Mathematical Reviews (MathSciNet): MR1245296
Zentralblatt MATH: 0788.60119
Digital Object Identifier: doi:10.1214/aop/1176989006
Project Euclid: euclid.aop/1176989006
[2] Dawson, D. A. (1977). The critical measure diffusion process. Z. Wahrsch. Verw. Gebiete 40 125–145.
Mathematical Reviews (MathSciNet): MR478374
Digital Object Identifier: doi:10.1007/BF00532877
[3] 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.
Mathematical Reviews (MathSciNet): MR1242575
[4] 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.
Mathematical Reviews (MathSciNet): MR1012498
Zentralblatt MATH: 0692.60063
Digital Object Identifier: doi:10.1007/BF00333147
[5] Dawson, D. A. and Perkins, E. A. (1991). Historical processes. Mem. Amer. Math. Soc. 93 iv+179.
Mathematical Reviews (MathSciNet): MR1079034
[6] Delmas, J.-F. (1999). Some properties of the range of super-Brownian motion. Probab. Theory Related Fields 114 505–547.
Mathematical Reviews (MathSciNet): MR1709279
Zentralblatt MATH: 0938.60082
Digital Object Identifier: doi:10.1007/s004400050233
[7] Dynkin, E. B. (1994). An Introduction to Branching Measure-Valued Processes. CRM Monograph Series 6. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR1280712
Zentralblatt MATH: 0824.60001
[8] Etheridge, A. M. (2000). An Introduction to Superprocesses. University Lecture Series 20. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR1779100
Zentralblatt MATH: 0971.60053
[9] Fleischmann, K. and Gärtner, J. (1986). Occupation time processes at a critical point. Math. Nachr. 125 275–290.
Mathematical Reviews (MathSciNet): MR847367
Zentralblatt MATH: 0596.60080
Digital Object Identifier: doi:10.1002/mana.19861250119
[10] Iscoe, I. (1988). On the supports of measure-valued critical branching Brownian motion. Ann. Probab. 16 200–221.
Mathematical Reviews (MathSciNet): MR920265
Digital Object Identifier: doi:10.1214/aop/1176991895
Project Euclid: euclid.aop/1176991895
[11] Kallenberg, O. (1986). Random Measures, 4th ed. Akademie-Verlag, Berlin.
Mathematical Reviews (MathSciNet): MR854102
[12] Kallenberg, O. (2001). Local hitting and conditioning in symmetric interval partitions. Stochastic Process. Appl. 94 241–270.
Mathematical Reviews (MathSciNet): MR1840832
Zentralblatt MATH: 1053.60086
Digital Object Identifier: doi:10.1016/S0304-4149(01)00086-2
[13] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). Springer, New York.
Mathematical Reviews (MathSciNet): MR1876169
[14] Kallenberg, O. (2003). Palm distributions and local approximation of regenerative processes. Probab. Theory Related Fields 125 1–41.
Mathematical Reviews (MathSciNet): MR1952454
Zentralblatt MATH: 1022.60043
Digital Object Identifier: doi:10.1007/s004400200218
[15] Kallenberg, O. (2005). Probabilistic Symmetries and Invariance Principles. Probability and Its Applications (New York). Springer, New York.
Mathematical Reviews (MathSciNet): MR2161313
Zentralblatt MATH: 1084.60003
[16] Kallenberg, O. (2007). Some problems of local hitting, scaling, and conditioning. Acta Appl. Math. 96 271–282.
Mathematical Reviews (MathSciNet): MR2327541
Digital Object Identifier: doi:10.1007/s10440-007-9103-4
[17] Kallenberg, O. (2007). Some local approximation properties of simple point processes. Probab. Theory Related Fields. To appear.
Mathematical Reviews (MathSciNet): MR2449123
Zentralblatt MATH: 1158.60015
Digital Object Identifier: doi:10.1007/s00440-007-0120-z
[18] Kingman, J. F. C. (1973). An intrinsic description of local time. J. London Math. Soc. (2) 6 725–731.
Mathematical Reviews (MathSciNet): MR319278
Zentralblatt MATH: 0283.60080
Digital Object Identifier: doi:10.1112/jlms/s2-6.4.725
[19] Klenke, A. (1998). Clustering and invariant measures for spatial branching models with infinite variance. Ann. Probab. 26 1057–1087.
Mathematical Reviews (MathSciNet): MR1634415
Zentralblatt MATH: 0938.60086
Digital Object Identifier: doi:10.1214/aop/1022855745
Project Euclid: euclid.aop/1022855745
[20] Le Gall, J.-F. (1986). Sur la saucisse de Wiener et les points multiples du mouvement brownien. Ann. Probab. 14 1219–1244.
Mathematical Reviews (MathSciNet): MR866344
Zentralblatt MATH: 0621.60083
Digital Object Identifier: doi:10.1214/aop/1176992364
Project Euclid: euclid.aop/1176992364
[21] Le Gall, J.-F. (1994). A lemma on super-Brownian motion with some applications. In The Dynkin Festschrift. Progr. Probab. 34 237–251. Birkhäuser, Boston, MA.
Mathematical Reviews (MathSciNet): MR1311723
Zentralblatt MATH: 0812.60066
[22] Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel.
[23] Mörters, P. (2001). The average density of super-Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 37 71–100.
Mathematical Reviews (MathSciNet): MR1815774
Zentralblatt MATH: 0978.60046
Digital Object Identifier: doi:10.1016/S0246-0203(00)01060-8
[24] Mörters, P. and Shieh, N.-R. (1999). Small scale limit theorems for the intersection local times of Brownian motion. Electron. J. Probab. 4 no. 9, 23 pp. (electronic).
Mathematical Reviews (MathSciNet): MR1690313
Zentralblatt MATH: 0937.60032
[25] Perkins, E. A. (1994). The strong Markov property of the support of super-Brownian motion. In The Dynkin Festschrift. Progr. Probab. 24 307–326. Birkhäuser, Boston, MA.
Mathematical Reviews (MathSciNet): MR1311727
Zentralblatt MATH: 0814.60046
[26] 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 125–329. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1915445
Zentralblatt MATH: 1020.60075
[27] Tribe, R. (1994). A representation for super Brownian motion. Stochastic Process. Appl. 51 207–219.
Mathematical Reviews (MathSciNet): MR1288289
Zentralblatt MATH: 0810.60075
Digital Object Identifier: doi:10.1016/0304-4149(94)90042-6
[28] Zähle, U. (1988). Self-similar random measures. I. Notion, carrying Hausdorff dimension, and hyperbolic distribution. Probab. Theory Related Fields 80 79–100.
Mathematical Reviews (MathSciNet): MR970472
Zentralblatt MATH: 0638.60064
Digital Object Identifier: doi:10.1007/BF00348753
