Branching random walks with random environments in time

Abstract

We consider a branching random walk on \mathbb{R} with a random environment in time (denoted by ξ). Let Z_n be the counting measure of particles of generation n, and let {\tilde{Z}}_n(t) be its Laplace transform. We show the convergence of the free energy n^-1log{\tilde{Z}}_n(t), large deviation principles, and central limit theorems for the sequence of measures {Z_n}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale {\tilde{Z}}_n(t)/\mathrm{E}\left[{\tilde{Z}}_n(t)|\xi \right].

Keywords

Branching random walk / random environment / large deviation / central limit theorem / moment

References

[1] Biggins J D. Martingale convergence in the branching random walk. J Appl Probab, 1977, 14: 25-37

[2] Biggins J D. Chernoff’s theorem in the branching random walk. J Appl Probab, 1977, 14: 630-636

[3] Biggins J D. The central limit theorem for the supercritical branching random walk, and related results. Stochastic Process Appl, 1990, 34: 255-274

[4] Biggins J D, Kyprianou A E. Measure change in multitype branching. Adv Appl Probab, 2004, 36(2): 544-581

[5] Chow Y, Teicher H. Probability Theory: Independence, Interchangeability, Martingales. Berlin: Springer-Verlag, 1995

[6] Comets F, Popov S, On multidimensional branching random walks in random environments. Ann Probab, 2007, 35: 68-114

[7] Comets F, Yoshida N. Branching random walks in space-time random environment: survival probability, global and local growth rates. J Theoret Probab, 2011, 24: 657-687

[8] Dembo A, Zeitouni O. Large Deviations Techniques and Applications. New York: Springer, 1998

[9] Guivarc’h Y, Liu Q. Propriétés asymptotiques des processus de branchement en environnement aléatoire. C R Acad Sci Paris, 2001, 332: 339-344

[10] Hu Y, Shi Z. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann Probab, 2009, 37(2): 742-789

[11] Huang C, Liu Q. Branching random walk with a random environment in time. Univ Bretagne-Sud, France, Preprint, 2013

[12] Kaplan N, Asmussen S. Branching random walks I. Stochastic Process Appl, 1976, 4: 1-13

[13] Kaplan N, Asmussen S. Branching random walks II. Stochastic Process Appl, 1976, 4: 15-31

[14] Kuhlbusch D. On weighted branching processes in random environment. Stochastic Process Appl, 2004, 109: 113-144

[15] Liang X, Liu Q. Mandelbrot’s cascades in random environments. Univ Bretagne-Sud, France, Preprint, 2013

[16] Liu Q. Branching random walks in random environment. In: Ji L, Liu K, Yang L, Yau S-T, eds. Proceedings of the 4th International Congress of Chinese Mathematicians, 2007, Vol II. 2007, 702-719

[17] Nakashima M. Almost sure central limit theorem for branching random walks in random environment. Ann Appl Probab, 2011, 21: 351-373

[18] Shi Z. Branching random walks. Saint-Flour’s Summer Course, 2012

[19] Yoshida N. Central limit theorem for random walk in random environment. Ann Appl Probab, 2008, 18: 1619-1635

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg