THE BRANCHING CHAIN WITH DRIFT IN SPACE-TIME RANDOM ENVIRONMENT (I): MODEL, MARKOV PROPERTY, MOMENTS

doi:10.1016/S0252-9602(10)60160-9

Abstract

Abstract:

There are three parts in this article. In Section 1, we establish the model of branching chain with drift in space-time random environment (BCDSTRE), i.e., the coupling of branching chain and random walk. In Section 2, we prove that any BCDSTRE must be a Markov chain in
time random environment when we consider the distribution of the particles in space as a random element. In Section 3, we calculate the first-order moments and the second-order moments of BCDSTRE.

Key words: branching chain with drift in space-time random environment, random branching generating function, local random transition generating function, random drift law, random branching law

CLC Number:

60J27

HU Di-He, HU Xiao-Yu. THE BRANCHING CHAIN WITH DRIFT IN SPACE-TIME RANDOM ENVIRONMENT (I): MODEL, MARKOV PROPERTY, MOMENTS[J].Acta mathematica scientia,Series B, 2010, 30(5): 1669-1678.

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References

[1] Sevastyanov B R. Theory of random branching processes. Uspekhi Mat Nank, 1951, 6: 47--99

[2] Sevastyanov B R. Branching Processes. Moscow: Nanka Press, 1971

[3] Athreya K B, Ney P E, Branching Processes. Berlin: Springer-Verlag, 1972

[4] Harris T E. The Theory of Branching Processes. Berlin: Springer-Verlag, 1963

[5] Kalinkin A V. Markov branching processes with interation. Russian Math Surveys, 2002, 57: 241--304

[6] Athreya K B, Kalin S. On branching processes with random environment, I: Extinction probabilities. Ann Math Statist, 1971, 42: 1499--1520

[7] Cohn H. A martingale approach to supercritical branching processes. Ann Probab, 1985, 13: 1179--1191

[8] Cohn H. On the growth of the multi type supercritical branching processes in a random environment. Ann Probab, 1989, 17: 1118--1123

[9] Solomon F. Random walks in a random environment. Ann Probab, 1975, 3: 1--31

[10] Kalikow S. Generalized random walks in random environments. Ann Probab, 1981, 9: 753--768

[11] Sinai Y G. The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab Appl, 1982, 27: 256--268

[12] Sznaitman A S. Slowdown estimates and central limit theorem for random environment. J Eur Math Soc, 2000, 2: 93--143

[13] Bérard J. The almost sure central limit theorem theorem for one-dimensional nearest-neighbour random walks is space-time random environment. J Appl Prob, 2004, 41: 83--92

[14] Rassoul-Agha F, Seppäläinen T. An almost sure invariance principle for random walks in a space-time random environment. Probab Th Rel Fields, 2005, 133(3): 299--314

[15] Zeitouni O. Random Walks in Random Environments. Lecture Notes in Mathematics 1837. Berlin: Spinger-Verlag, 2004: 189--312

[16] Nawrotzki K. Discrete open system of Markov chains in arandom environment, I, II. J Inform Process, Cybernet, 1981, 17: 569--599; 1982, 18: 83--98

[17] Cogburn R. Markov chains in random environments: the case of Markovian environments. Ann Probab, 1980, 8: 908--916

[18] Cogburn R. The ergodic of Markov chains in random environments. Z Wahrach Verw, Gebiete, 1984, 66: 109--128

[19] Cogburn R. On the central limit theorem for Markov chains in random environments. Ann Probab, 1991, 19: 587--604

[20] Orey S. Markov chains with stochastically stationary transition probabilities. Ann Probab, 1991, 19: 907--928

[21] Yang G, Hu D. Model of Markov chains in space-time random environments. Wuhan Univ J Natural Sci, 2007, 12(2): 225--229

[22] Hu D. The construction of Markov processes in random environments and equivalence theorems. Science in China (Series A), 2004, 47: 481--496

[23] Hu D. The exestence and uniqueness of q-processes in random environments. Science in China (Series A), 2004, 47: 641--658

[24] Hu D. Infinitely dimensional contral Markov branching chains in random environments. Science in China (Series A), 2005, 48: 27--53

[25] Hu D, Hu X. On Markov chains in space-time random environments. Acta Math Sci, 2009, 29B(1): 1--10

THE BRANCHING CHAIN WITH DRIFT IN SPACE-TIME RANDOM ENVIRONMENT (I): MODEL, MARKOV PROPERTY, MOMENTS

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