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\(L \log L\) criterion for a class of superdiffusions. (English) Zbl 1175.60077

The martingale change of measure method developed by R. Lyons, R. Pemantle and Y. Peres [Ann. Probab. 23, No. 3, 1125–1138 (1995; Zbl 0840.60077)] for an alternative proof of the Kesten-Stigum theorem for supercritical one-type branching processes and later extended by various other authors to more general branching processes is extended here to a class of super-diffusions. A main tool is a spine decomposition of super-diffusions.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F15 Strong limit theorems
60J25 Continuous-time Markov processes on general state spaces

Citations:

Zbl 0840.60077
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References:

[1] Asmussen, S. and Hering, H. (1976). Strong limit theorems for general supercritical branching processes with applications to branching diffusions. Z.Wahrscheinlichkeitsth. 36 , 195–212. · Zbl 0325.60081 · doi:10.1007/BF00532545
[2] Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. Appl. Prob. 36 , 544–581. · Zbl 1056.60082 · doi:10.1239/aap/1086957585
[3] Durrett, R. (1996). Probability Theory and Examples , 2nd edn. Duxbury Press, Belmont, CA. · Zbl 0709.60002
[4] Dynkin, E. B. (1993). Superprocesses and partial differential equations. Ann. Prob. 21 , 1185–1262. · Zbl 0806.60066 · doi:10.1214/aop/1176989116
[5] Engländer, J. and Kyprianou, A. E. (2004). Local extinction versus local exponential growth for spatial branching processes. Ann. Prob. 32 , 78–99. · Zbl 1056.60083 · doi:10.1214/aop/1078415829
[6] Evans, S. N. (1992). Two representations of a conditioned superprocess. Proc. R. Soc. Edinburgh A 123 , 959–971. · Zbl 0784.60052 · doi:10.1017/S0308210500029619
[7] Harris, S. C. and Roberts, M. (2008). Measure changes with extinction. Statist. Prob. Lett. 79 , 1129–1133. · Zbl 1163.60309 · doi:10.1016/j.spl.2008.12.025
[8] Kesten, H. and Stigum, B. P. (1966). A limit theorem for multidimensional Galton–Watson process. Ann. Math. Statist. 37 , 1211–1223. · Zbl 0203.17401 · doi:10.1214/aoms/1177699266
[9] Kim, P. and Song, R. (2008). Intrinsic ultracontractivity of non-symmetric diffusion semigroups in bounded domains. Tohoku Math. J. 60 , 527–547. · Zbl 1167.60017 · doi:10.2748/tmj/1232376165
[10] Kim, P. and Song, R. (2008). Intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and potentials. Ann. Prob. 36 , 1904–1945. · Zbl 1175.47039 · doi:10.1214/07-AOP381
[11] Kim, P. and Song, R. (2009). Intrinsic ultracontractivity for non-symmetric Lévy processes. Forum Math. 21 , 43–66. · Zbl 1175.47040 · doi:10.1515/FORUM.2009.003
[12] Kurtz, T., Lyons, R., Pemantle, R. and Peres, Y. (1997). A conceptual proof of the Kesten–Sigum theorem for multi-type branching processes. In Classical and Modern Branching processes (Minneapolis, 1994; IMA Vol. Math. Appl. 84 ), eds K. B. Athreya and P. Jagers, Springer, New York, pp. 181–186. · Zbl 0868.60068
[13] Lyons, R. (1997). A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching processes (Minneapolis, 1994; IMA Vol. Math. Appl. 84 ), eds K. B. Athreya and P. Jagers, Springer, New York, pp. 217–222. · Zbl 0897.60086
[14] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of \(L\log L\) criteria for mean behavior of branching processes. Ann. Prob. 23 , 1125–1138. · Zbl 0840.60077 · doi:10.1214/aop/1176988176
[15] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions (Camb. Stud. Adv. Math. 68 ). Cambridge University Press. · Zbl 0973.60001
[16] Schaeffer, H. H. (1974). Banach Lattices and Positive Operators. Springer, New York. · Zbl 0296.47023
[17] Sharpe, M. (1988). General Theory of Markov Processes (Pure Appl. Math. 133 ). Academic Press, Boston, MA. · Zbl 0649.60079
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