Asymptotic Behavior of Certain Branching Processes

Loading...
Thumbnail Image

Date

2019

Journal Title

Journal ISSN

Volume Title

Repository Usage Stats

381
views
452
downloads

Abstract

This dissertation examines the asymptotic behavior of three branching processes. The first is a branching process with selection; the selection is dictated by a fitness function which is the sum of a linear part and a periodic part. It is shown that the system has an asymptotic speed and that there is a stationary distribution in an appropriate moving frame. This is done through an examination of tightness of the process and application of an ergodic theorem. The second process studied is a branching process with selection driven by a symmetric function with a single local maximum at the origin and which monotonically decreases away from the origin. For this process, a large particle limit of the system is proven and related to the solution to a free boundary partial differential equation. Finally, a branching process is studied in which the branch rate of particles is a function of the empirical measure. Weak convergence to the solution of a non-local partial differential equation is proven. Tightness is proven first, and then the limit object is characterized by its behavior when applied to test functions.

Department

Description

Provenance

Subjects

Citation

Citation

Beckman, Erin Melissa (2019). Asymptotic Behavior of Certain Branching Processes. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/18756.

Collections


Except where otherwise noted, student scholarship that was shared on DukeSpace after 2009 is made available to the public under a Creative Commons Attribution / Non-commercial / No derivatives (CC-BY-NC-ND) license. All rights in student work shared on DukeSpace before 2009 remain with the author and/or their designee, whose permission may be required for reuse.