Publication Date

2018-07-23

Availability

Open access

Embargo Period

2018-07-23

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PHD)

Department

Mathematics (Arts and Sciences)

Date of Defense

2018-06-13

First Committee Member

Ilie Grigorescu

Second Committee Member

Robert Stephen Cantrell

Third Committee Member

Subramanian Ramakrishnan

Fourth Committee Member

Burton J. Rosenberg

Abstract

We develop the continuous time version of the particle model studied in [13]. The evolution is described by a pure jump, continuous time Markov process on the space of words of length L with a size N alphabet. Words change randomly in search of a preferred state, here the vector zero. In genome population models, this is the genome presenting selection advantage [21]; in cancer development, it is a state of a damaged gene by deleterious mutations, and in epidemiological models the number of infected individuals in the population. In the last two models, the characters in the preferred word have a probability of returning among ordinary states. It will turn out to be essential that depends on the configuration, leading to an interacting particle system. We investigate the scaling limit of the empirical measure, and study several types of random perturbations, together with applications. Chapter 1 presents the mathematical model. Chapter 2 proves the Fluid Limit, i.e. a Law of Large Numbers for a (random) dynamical system; Chapter 3 determines the Fluctuation near Equilibria, a fine scale (second order approximation) result; and Chapter 4, titled Generalized Logistic Equation with Noise, explores the relationship between quasi-stationarity and stable equilibria, a random perturbation question.

Keywords

Stochastic Process; Genetics; Applied Mathematics; Fluid Limit; Stability

Share

COinS