The Limit Behavior of Dual Markov Branching Processes

A dual Markov branching process (DMBP) is by definition a Siegmund's predual of some Markov branching process (MBP). Such a process does exist and is uniquely determined by the so-called dual-branching property. Its q-matrix Q is derived and proved to be regular and monotone. Several equivalent definitions for a DMBP are given. The criteria for transience, positive recurrence, strong ergodicity, and the Feller property are established. The invariant distributions are given by a clear formulation with a geometric limit law.

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Skeletal stochastic differential equations for superprocesses

Journal of Applied Probability ◽

10.1017/jpr.2020.53 ◽

2020 ◽

Vol 57 (4) ◽

pp. 1111-1134

Author(s):

Dorottya Fekete ◽

Joaquin Fontbona ◽

Andreas E. Kyprianou

Keyword(s):

Branching Process ◽

Branching Processes ◽

Subcritical Case ◽

Markov Branching Process ◽

Continuous State ◽

Common Framework ◽

Infinite Line ◽

Current Article ◽

Lines Of Descent ◽

Skeletal Representation

AbstractIt is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

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A note on Markov branching processes

Advances in Applied Probability ◽

10.1017/s0001867800015081 ◽

1985 ◽

Vol 17 (02) ◽

pp. 463-464

Author(s):

Fred M. Hoppe

Keyword(s):

Laplace Transform ◽

Simple Proof ◽

Branching Process ◽

Branching Processes ◽

Markov Branching Process ◽

The Laplace Transform

We present a simple proof of Zolotarev’s representation for the Laplace transform of the normalized limit of a Markov branching process and relate it to the Harris representation.

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Extinction Times in Multitype Markov Branching Processes

Journal of Applied Probability ◽

10.1017/s0021900200005374 ◽

2009 ◽

Vol 46 (01) ◽

pp. 296-307 ◽

Cited By ~ 3

Author(s):

Dominik Heinzmann

Keyword(s):

Limit Theorem ◽

Continuous Time ◽

Branching Process ◽

Branching Processes ◽

Markov Branching Process ◽

Time To Extinction

In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.

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Uniqueness and Decay Properties of Markov Branching Processes with Disasters

Journal of Applied Probability ◽

10.1017/s0021900200011554 ◽

2014 ◽

Vol 51 (03) ◽

pp. 613-624 ◽

Cited By ~ 1

Author(s):

Anyue Chen ◽

Kai Wang Ng ◽

Hanjun Zhang

Keyword(s):

Branching Process ◽

Invariant Measures ◽

Branching Processes ◽

Decay Parameter ◽

Markov Branching Process ◽

Decay Properties ◽

Invariant Vectors ◽

Quasistationary Distributions

In this paper we discuss the decay properties of Markov branching processes with disasters, including the decay parameter, invariant measures, and quasistationary distributions. After showing that the corresponding q-matrix Q is always regular and, thus, that the Feller minimal Q-process is honest, we obtain the exact value of the decay parameter λ C . We show that the decay parameter can be easily expressed explicitly. We further show that the Markov branching process with disaster is always λ C -positive. The invariant vectors, the invariant measures, and the quasidistributions are given explicitly.

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Extinction Times in Multitype Markov Branching Processes

Journal of Applied Probability ◽

10.1239/jap/1238592131 ◽

2009 ◽

Vol 46 (1) ◽

pp. 296-307 ◽

Cited By ~ 8

Author(s):

Dominik Heinzmann

Keyword(s):

Limit Theorem ◽

Continuous Time ◽

Branching Process ◽

Branching Processes ◽

Markov Branching Process ◽

Time To Extinction

In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.

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A note on Markov branching processes

Advances in Applied Probability ◽

10.2307/1427152 ◽

1985 ◽

Vol 17 (2) ◽

pp. 463-464 ◽

Cited By ~ 1

Author(s):

Fred M. Hoppe

Keyword(s):

Laplace Transform ◽

Simple Proof ◽

Branching Process ◽

Branching Processes ◽

Markov Branching Process ◽

The Laplace Transform

We present a simple proof of Zolotarev’s representation for the Laplace transform of the normalized limit of a Markov branching process and relate it to the Harris representation.

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A note on the multitype Markov branching process

Journal of Applied Probability ◽

10.2307/3214420 ◽

1989 ◽

Vol 26 (3) ◽

pp. 631-636 ◽

Cited By ~ 1

Author(s):

V. G. Gadag

Keyword(s):

Central Limit ◽

Limit Theorems ◽

Branching Process ◽

Central Limit Theorems ◽

Asymptotic Results ◽

Original Process ◽

Markov Branching Process ◽

Lines Of Descent ◽

Branching Property

We consider a supercritical, p-dimensional Markov branching process (MBP). Based on the finite and the infinite lines of descent of particles of this p-dimensional MBP, we construct an associated 2p-dimensional process. We show that such a process is a 2p-dimensional, supercritical MBP. This 2p-dimensional process retains the branching property when conditioned on the sets of extinction and non-extinction. Asymptotic results and central limit theorems for the associated process and the original process are established by using the results of Gadag and Rajarshi (1987).

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Upper Deviations for Split Times of Branching Processes

Journal of Applied Probability ◽

10.1239/jap/1354716662 ◽

2012 ◽

Vol 49 (4) ◽

pp. 1134-1143

Author(s):

Hamed Amini ◽

Marc Lelarge

Keyword(s):

Continuous Time ◽

Branching Process ◽

Branching Processes ◽

Deviation Probability ◽

Markov Branching Process ◽

Split Time

Upper deviation results are obtained for the split time of a supercritical continuous-time Markov branching process. More precisely, we establish the existence of logarithmic limits for the likelihood that the split times of the process are greater than an identified value and determine an expression for the limiting quantity. We also give an estimation for the lower deviation probability of the split times, which shows that the scaling is completely different from the upper deviations.

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Limit Properties of Transition Functions of Continuous-Time Markov Branching Processes

International Journal of Stochastic Analysis ◽

10.1155/2014/409345 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10

Author(s):

Azam A. Imomov

Keyword(s):

Continuous Time ◽

Limit Theorems ◽

Branching Process ◽

Branching Processes ◽

Local Limit ◽

Markov Branching Process ◽

Transition Functions ◽

Basic Lemma ◽

Invariant Properties ◽

Local Limit Theorems

Consider the Markov Branching Process with continuous time. Our focus is on the limit properties of transition functions of this process. Using differential analogue of the Basic Lemma we prove local limit theorems for all cases and observe invariant properties of considering process.

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