A spine approach to branching diffusions with applications to L p -convergence of martingales

Lecture Notes in Mathematics

Volumn 1979, Issue , 2009, Pages 281-330

  Hardy, Robert ^a   Harris, Simon C ^a  

^a University of Bath   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 68749086062     PISSN: 00758434     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-642-01763-6_11     Document Type: Article

References (49)

1. S. Asmussen and H. Hering, Strong limit theorems for general supercritical branching processes with applications to branching diffusions, Z. Wahrschein-lichkeitstheorie und Verw. Gebiete 36 (1976), no. 3, 195-212.
2. K. B. Athreya, Change of measures for Markov chains and the L log L theorem for branching processes, Bernoulli 6 (2000), no. 2, 323-338.5
3. J. Bertoin, Random fragmentation and coagulation processes, Cambridge University Press, 2006.
4. J. D. Biggins and A. E. Kyprianou, Measure change in multitype branching, Adv. in Appl. Probab. 36 (2004), no. 2, 544-581.
5. A. Champneys, S. Harris, J. Toland, J. Warren, and D. Williams, Algebra, analysis and probability for a coupled system of reaction-diffusion equations, Philosophical Transactions of the Royal Society of London 350 (1995), 69-112.
6. B. Chauvin, Arbres et processus de Bellman-Harris, Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 2, 209-232.
7. B. Chauvin and A. Rouault, Boltzmann-Gibbs weights in the branching random walk, Classical and modern branching processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 41-50.
8. B. Chauvin, Product martingales and stopping lines for branching Brownian motion, Ann. Probab. 19 (1991), no. 3, 1195-1205.
9. B. Chauvin and A. Rouault, KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees, Probab. Theory Related Fields 80 (1988), no. 2, 299-314.
10. B. Chauvin, A. Rouault, and A. Wakolbinger, Growing conditioned trees, Stochastic Process. Appl. 39 (1991), no. 1, 117-130.
11. R. Durrett, Probability: Theory and examples, 2nd ed., Duxbury Press, 1996.
12. J. Engländer and A. E. Kyprianou, Local extinction versus local exponential growth for spatial branching processes, Ann. Probab. 32 (2004), no. 1A, 78-99.
13. J. Engländer, Branching diffusions, superdiffusions and random media, Probab. Surveys Vol. 4 (2007) 303-364.
14. J. Engländer, S. C. Harris, and A. E. Kyprianou, Laws of Large numbers for spatial branching processes, Annales de l'Institut Henri Poincaré (B) Probability and Statistics, (2009), to appear.
15. J. Geiger, Size-biased and conditioned random splitting trees, Stochastic Process. Appl. 65 (1996), no. 2, 187-207.
16. J. Geiger, Elementary new proofs of classical limit theorems for Galton-Watson processes, J. Appl. Probab. 36 (1999), no. 2, 301-309.
17. J. Geiger, Poisson point process limits in size-biased Galton-Watson trees, Electron. J. Probab. 5 (2000), no. 17, 12 pp. (electronic).
18. J. Geiger and L. Kauffmann, The shape of large Galton-Watson trees with possibly infinite variance, Random Structures Algorithms 25 (2004), no. 3, 311-335.
19. H. O. Georgii and E. Baake, Supercritical multitype branching processes: the ancestral types of typical individuals, Adv. in Appl. Probab. 35 (2003), no. 4, 1090-1110.
20. Y. Git, J. W. Harris, and S. C. Harris, Exponential growth rates in a typed branching diffusion, Ann. App. Probab., 17 (2007), no. 2, 609-653. doi:10.1214/105051606000000853
21. R. Hardy, Branching diffusions, Ph.D. thesis, University of Bath Department of Mathematical Sciences, 2004.
22. R. Hardy and S. C. Harris, Some path large deviation results for a branching diffusion, (2007), submitted.
23. R. Hardy and S. C. Harris, A conceptual approach to a path result for branching Brownian motion, Stoch. Proc. and Applic., 116 (2006), no. 12, 1992-2013. doi:10.1016/j.spa.2006.05.010
24. R. Hardy and S. C. Harris, A new formulation of the spine approach for branching diffusions, (2006). arXiv:math.PR/0611054
25. p-convergence of branching-diffusion martingales, (2006). arXiv:math.PR/0611056
26. J. W. Harris, S. C. Harris, and A. E. Kyprianou, Further probabilistic analysis of the Fisher-Kolmogorov-Petrovskii-Piscounov equation: one-sided travelling waves, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 1, 125-145.
27. J. W. Harris and S. C. Harris, Branching Brownian motion with an inhomo-geneous breeding potential, Ann. Inst. H. Poincaré Probab. Statist. (2008), to appear.
28. S. C. Harris and D. Williams, Large deviations and martingales for a typed branching diffusion. I, Astérisque (1996), no. 236, 133-154, Hommage a P. A. Meyer et J. Neveu.
29. S. C. Harris, Travelling-waves for the FKPP equation via probabilistic arguments, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 3, 503-517.
30. S. C. Harris, Convergence of a "Gibbs-Boltzmann" random measure for a typed branching diffusion, Séminaire de Probabilités, XXXIV, Lecture Notes in Math., vol. 1729, Springer, Berlin, 2000, pp. 239-256.
31. S. C. Harris, R. Knobloch, and A.E. Kyprianou, Strong Law of Large Numbers for Fragmentation Processes, (2008) arXiv:0809.2958v1, submitted.
32. T. E. Harris, The theory of branching processes, Dover ed., Dover, 1989.
33. Y. Hu and Z. Shi, Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees, (2008) Ann. App. Probab., to appear.
34. A. M. Iksanov, Elementary fixed points of the BRW smoothing transforms with infinite number of summands, Stochastic Process. Appl. 114 (2004), no. 1, 27-50.
35. O. Kallenberg, Foundations of modern probability, Springer-Verlag, 2002.
36. H. Kesten and B. P. Stigum, Additional limit theorems for indecomposable multidimensional Galton-Watson processes, Ann. Math. Stat. 37 (1966), 1463-1481.
37. H. Kesten and B. P. Stigum, A limit theorem for multidimensional Galton-Watson processes, Ann. Math. Stat. 37 (1966), 1211-1223.
38. H. Kesten and B. P. Stigum, Limit theorems for decomposable multi-dimensional Galton-Watson processes, J. Math. Anal. Applic. 17 (1967), 309-338.
39. E. Kreyszig, Introductory functional analysis with applications, John Wiley and Sons, 1989.
40. A. E. Kyprianou, Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 1, 53-72.
41. T. Kurtz, R. Lyons, R. Pemantle, and Y. Peres, A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes, Classical and modern branching processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 181-185.
42. Q. Liu and A. Rouault, On two measures defined on the boundary of a branching tree, Classical and modern branching processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 187-201.
43. R. Lyons, A simple path to Biggins' martingale convergence for branching random walk, Classical and modern branching processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 217-221.
44. R. Lyons, R. Pemantle, and Y. Peres, Conceptual proofs of L log L criteria for mean behavior of branching processes, Ann. Probab. 23 (1995), no. 3, 1125-1138.
45. J. Neveu, Arbres et processus de Galton-Watson, Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 2, 199-207.
46. J. Neveu, Multiplicative martingales for spatial branching processes, Seminar on Stochastic Processes (E. Çinlar, K.L.Chung, and R.K.Getoor, eds.), Birkhäuser, 1987, pp. 223-241.
47. P. Olofsson, The x log x condition for general branching processes, J. Appl. Probab. 35 (1998), no. 3, 537-544.
48. E. Seneta, Non-negative matrices and Markov chains, Springer-Verlag, 1981.
49. E. C. Waymire and S. C. Williams, A general decomposition theory for random cascades, Bull. Amer. Math. Soc. (N.S.) 31 (1994), no. 2, 216-222.