Branching Processes with Immigration and Related Topics

Li, Zenghu

doi:10.1007/s11464-005-0027-x

Cited by 25 publications

(20 citation statements)

References 69 publications

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“…This implies that (w n , n ≥ 0) increases to a non-negative function w ∞ . By monotone convergence theorem (for H µ s ψ 0 (w(r))1 {wn(r)>0} dr and the integral with φ) and dominated convergence theorem (for H µ s ψ 0 (w(r))1 {wn(r)≤0} dr), we deduce from (11), that w ∞ solves w(s) = 0 for s > H µ and Notice that w ∞ (s) ∈ [0, ∞] and the two sides of the previous equality may be infinite. Thanks to Proposition 2.1, and since ψ 0 − φ is a branching mechanism (see Remark 3.2), there exists a unique locally bounded non-negative solution of (12), which we shall callw.…”

Section: 2mentioning

confidence: 90%

Changing the branching mechanism of a continuous state branching process using immigration

Abraham¹,

Delmas²

2009

Ann. Inst. H. Poincaré Probab. Statist.

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International audienceWe construct a continuous state branching process with immigration (CBI) whose immigration depends on the CBI itself and we recover a continuous state branching process (CB). This provides a dual construction of the pruning at nodes of CB introduced by the authors in a previous paper. This construction is a natural way to model neutral mutation. Using exponential formula, we compute the probability of extinction of the original type population in a critical or sub-critical quadratic branching, conditionally on the non extinction of the total population

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Section: 2mentioning

confidence: 90%

Changing the branching mechanism of a continuous state branching process using immigration

Abraham¹,

Delmas²

2009

Ann. Inst. H. Poincaré Probab. Statist.

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“…Let us therefore first consider separately the branching structure. In practice the evolution of the number of nodes (i.e., the amount of messages disseminated in the network) is governed by continuous state branching processes (CSBPs), introduced by Jiřina [1958] and studied by various researchers during the past decades [see for example Lamperti, 1967a,b, Feller, 1951, Grey, 1974, Silverstein, 1969, Bingham, 1976, Le Gall, 1999, Li, 2006, Kale and Deshmukh, 1992, Li, 2009, Caballero et al, 2009, Li, 2012a, Kashikar and Deshmukh, 2012. In order to understand what a CSBP is, we start by describing a related discrete time and discrete state space model: the so-called Galton-Watson process.…”

Section: Mathematical Background: Super Brownian Motionmentioning

confidence: 99%

Superprocesses as Models for Information Dissemination in the Future Internet

Sacerdote

Garetto

Polito

et al. 2014

Mathematical Models and Methods for Planet Earth

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Future Internet will be composed by a tremendous number of potentially interconnected people and devices, offering a variety of services, applications and communication opportunities. In particular, short-range wireless communications, which are available on almost all portable devices, will enable the formation of the largest cloud of interconnected, smart computing devices mankind has ever dreamed about: the Proximate Internet. In this paper, we consider superprocesses, more specifically super Brownian motion, as a suitable mathematical model to analyse a basic problem of information dissemination arising in the context of Proximate Internet. The proposed model provides a promising analytical framework to both study theoretical properties related to the information dissemination process and to devise efficient and reliable simulation schemes for very large systems.

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“…The concept of two-dimensional CB-processes was introduced by Watanabe (1969) and the corresponding processes with immigration have been systematically studied by several authors in the measure-valued setting; see, e.g., Li (2006b) for the survey on the topic. Two-dimensional CBI-processes are included in a wide class of so-called affine processes and many interesting applications of the class of processes have been found in mathematical finance; see Duffie et al (2003).…”

Section: Introductionmentioning

confidence: 99%