Branching Lévy Processes with Inhomogeneous Breeding Potentials

Sergey Bocharov

Research output: ThesisDoctoral Thesis

Abstract

The object of study in this thesis is a number of different models of branching Levy processes in inhomogeneous breeding potential. We employ some widely-used spine techniques to investigate various features of these models for their subsequent comparison.
The thesis is divided into 5 chapters.
In the first chapter we introduce the general framework for branching Markov processes
within which we are going to present all our results.
In the second chapter we consider a branching Brownian motion in the potential β|·|p, β> 0, p ≥0. We give a new proof of the result about the critical value of p for the explosion time of the population. The main advantage of the new proof is that it can be easily generalised to other models.
The third chapter is devoted to continuous-time branching random walks in the potential β|·|p, β> 0, p ≥0. We give results about the explosion time and the right most particle behaviour comparing them with the known results for the branching Brownian motion.
In the fourth chapter we look at general branching Levy processes in the potential β|·|p, β> 0, p ≥0. Subject to certain assumptions we prove some results about the explosion time and the rightmost particle. We exhibit how the corresponding results for the branching Brownian motion and and the branching random walk fit into the general structure.
The last chapter considers a branching Brownian motion with branching taking place at the origin on the local time scale. We present some results about the population dynamics and the right most particle behaviour. We also prove the Strong Law of Large Numbers for this model.
LanguageEnglish
QualificationPh.D.
Awarding Institution
  • University of Bath
Supervisors/Advisors
  • Harris, Simon, Supervisor
Award date29 Sep 2012
StatusUnpublished - Jul 2012

Fingerprint

Branching process
Lévy Process
Branching Brownian Motion
Explosion
Branching Random Walk
Markov Branching Process
Continuous Time Random Walk
Strong law of large numbers
Spine
Local Time
Population Dynamics
Model
Critical value
Branching
Time Scales

Cite this

Branching Lévy Processes with Inhomogeneous Breeding Potentials. / Bocharov, Sergey.

2012.

Research output: ThesisDoctoral Thesis

Bocharov, S 2012, 'Branching Lévy Processes with Inhomogeneous Breeding Potentials', Ph.D., University of Bath.
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N2 - The object of study in this thesis is a number of different models of branching Levy processes in inhomogeneous breeding potential. We employ some widely-used spine techniques to investigate various features of these models for their subsequent comparison.The thesis is divided into 5 chapters.In the first chapter we introduce the general framework for branching Markov processeswithin which we are going to present all our results.In the second chapter we consider a branching Brownian motion in the potential β|·|p, β> 0, p ≥0. We give a new proof of the result about the critical value of p for the explosion time of the population. The main advantage of the new proof is that it can be easily generalised to other models.The third chapter is devoted to continuous-time branching random walks in the potential β|·|p, β> 0, p ≥0. We give results about the explosion time and the right most particle behaviour comparing them with the known results for the branching Brownian motion.In the fourth chapter we look at general branching Levy processes in the potential β|·|p, β> 0, p ≥0. Subject to certain assumptions we prove some results about the explosion time and the rightmost particle. We exhibit how the corresponding results for the branching Brownian motion and and the branching random walk fit into the general structure.The last chapter considers a branching Brownian motion with branching taking place at the origin on the local time scale. We present some results about the population dynamics and the right most particle behaviour. We also prove the Strong Law of Large Numbers for this model.

AB - The object of study in this thesis is a number of different models of branching Levy processes in inhomogeneous breeding potential. We employ some widely-used spine techniques to investigate various features of these models for their subsequent comparison.The thesis is divided into 5 chapters.In the first chapter we introduce the general framework for branching Markov processeswithin which we are going to present all our results.In the second chapter we consider a branching Brownian motion in the potential β|·|p, β> 0, p ≥0. We give a new proof of the result about the critical value of p for the explosion time of the population. The main advantage of the new proof is that it can be easily generalised to other models.The third chapter is devoted to continuous-time branching random walks in the potential β|·|p, β> 0, p ≥0. We give results about the explosion time and the right most particle behaviour comparing them with the known results for the branching Brownian motion.In the fourth chapter we look at general branching Levy processes in the potential β|·|p, β> 0, p ≥0. Subject to certain assumptions we prove some results about the explosion time and the rightmost particle. We exhibit how the corresponding results for the branching Brownian motion and and the branching random walk fit into the general structure.The last chapter considers a branching Brownian motion with branching taking place at the origin on the local time scale. We present some results about the population dynamics and the right most particle behaviour. We also prove the Strong Law of Large Numbers for this model.

M3 - Doctoral Thesis

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