On the theory of birth, death and diffusion processes

Abstract

Several authors have recently discussed the asymptotic properties of stochastic populations which diffuse randomly throughout a given region. Sevast'yanov ([8], [9]) has investigated the extinction probability of a Markovian population in a compact region with an absorbing boundary, his analysis being in terms of “generation times”. Adke and Moyal have considered the spatial dispersion of a population which multiplies according to a simple time-dependent birth-and-death process and undergoes Gaussian diffusion on the real line ([2] and [3]) or on a finite interval with reflecting boundaries [1]. A serious limitation in Adke and Moyal's asymptotic results is that they are conditional upon a finite number of survivors. Moyal [7] has also obtained some basic formulae for a Markovian population diffusing through a general space.