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Small positive values for supercritical branching processes in random environment

Vincent Bansaye, Christian Böinghoff (2014)

Annales de l'I.H.P. Probabilités et statistiques

Branching Processes in Random Environment (BPREs) ( Z n : n ≥ 0 ) are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of ℙ ( 1 ≤ Z n ≤ k | Z 0 = i ) , k , i ∈ ℕ as n → ∞ . More precisely, we characterize the exponential...

Some properties of superprocesses under a stochastic flow

Kijung Lee, Carl Mueller, Jie Xiong (2009)

Annales de l'I.H.P. Probabilités et statistiques

For a superprocess under a stochastic flow in one dimension, we prove that it has a density with respect to the Lebesgue measure. A stochastic partial differential equation is derived for the density. The regularity of the solution is then proved by using Krylov’s Lp-theory for linear SPDE.

Some Theoretical Results on the Progeny of a Bisexual Galton-Watson Branching Process

González, M., Molina, M. (1997)

Serdica Mathematical Journal

A Superadditive Bisexual Galton-Watson Branching Process is considered and the total number of mating units, females and males, until the n-th generation, are studied. In particular some results about the stochastic monotony, probability generating functions and moments are obtained. Finally, the limit behaviour of those variables suitably normed is investigated.

Spectral gap and convex concentration inequalities for birth–death processes

Wei Liu, Yutao Ma (2009)

Annales de l'I.H.P. Probabilités et statistiques

In this paper, we consider a birth–death process with generator ℒ and reversible invariant probabilityπ. Given an increasing function ρ and the associated Lipschitz norm ‖⋅‖Lip(ρ), we find an explicit formula for ∥ ( - ℒ ) - 1 ∥ Lip ( ρ ) . As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of ℒ inL2(π). Moreover, by Lyons–Zheng’s forward-backward martingale decomposition theorem, we get convex concentration inequalities for additive functionals of birth–death...

Strong law of large numbers for branching diffusions

János Engländer, Simon C. Harris, Andreas E. Kyprianou (2010)

Annales de l'I.H.P. Probabilités et statistiques

Let X be the branching particle diffusion corresponding to the operator Lu+β(u2−u) on D⊆ℝd (where β≥0 and β≢0). Let λc denote the generalized principal eigenvalue for the operator L+β on D and assume that it is finite. When λc>0 and L+β−λc satisfies certain spectral theoretical conditions, we prove that the random measure exp{−λct}Xt converges almost surely in the vague topology as t tends to infinity. This result is motivated by a cluster of articles due to Asmussen and Hering dating from...

Strong solutions for stochastic differential equations with jumps

Zenghu Li, Leonid Mytnik (2011)

Annales de l'I.H.P. Probabilités et statistiques

General stochastic equations with jumps are studied. We provide criteria for the uniqueness and existence of strong solutions under non-Lipschitz conditions of Yamada–Watanabe type. The results are applied to stochastic equations driven by spectrally positive Lévy processes.

Supercritical super-brownian motion with a general branching mechanism and travelling waves

A. E. Kyprianou, R.-L. Liu, A. Murillo-Salas, Y.-X. Ren (2012)

Annales de l'I.H.P. Probabilités et statistiques

We offer a probabilistic treatment of the classical problem of existence, uniqueness and asymptotics of monotone solutions to the travelling wave equation associated to the parabolic semi-group equation of a super-Brownian motion with a general branching mechanism. Whilst we are strongly guided by the reasoning in Kyprianou (Ann. Inst. Henri Poincaré Probab. Stat.40 (2004) 53–72) for branching Brownian motion, the current paper offers a number of new insights. Our analysis incorporates the role...

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