Some Time Changed Birth Death Processes and Poisson Random Fields

Abstract

In this thesis, we study some time-changed variants of the birth-death process. First, newlinewe consider a time-changed path integral of the homogeneous birth-death process, where newlinethe time changes according to an inverse stable subordinator. In a linear case, the limiting behavior of this integral process has been studied. newlineWe introduce and study a time-changed variant of the linear birth-death process under newlineimmigration effects. First, we consider the immigration effect at every possible states. Then,westudy the immigration effect in time-changed linear birth-death process where immigration occurs only if the population goes extinct. Also, some particular cases of both these processes are studied. newlineWestudy afractional birth-death process with state dependent birth and death rates. It is newlinedefined using a system of fractional differential equations that governs its state probabilities. newlineWe obtain the closed form expressions for its transient probabilities. In this way, we obtain the unknown transient probabilities for the classical birth-death process introduced by Feller newline(1939). newlineWeconsider a generalized birth-death process (GBDP) in which at any instance there can newlinebe multiple births or deaths with positive probability. In the case of constant birth and death rates in GBDP, the explicit forms of its state probabilities, joint probability mass functions of population size with cumulative births and cumulative deaths, and their marginal probability mass functions are obtained. Also, we consider a generalized birth-death process whose state space is a finite subset of a q-dimensional lattice. For a GBDP on two dimensional finite grid, we obtain a sufficient and necessary condition for the vertical and horizontal transition probability matrices to commute. Further, we extend these results to the case of q-dimensional finite grid. newlineLater, we consider a fractional Poisson random field on positive plane. We use the Adomian decomposition method to obtain a closed form expression for its probability mass (Full abstract uploaded)