Pure birth processes in a random medium (Q1823557)

Suppose that \(\lambda_ n\), \(\lambda_ n\in (0,1)\), are independent random variables on a p.s. (probability space) (\(\Omega\),F,P) and take values on the inverval (0,1), and \(\Omega\) consists of all possible states of a random medium. For fixed \(\omega\in \Omega\), this paper assumes \(x_ t(\omega,\cdot)\) is a Markov chain with continuous time and intensities of transition from a point n to a point \(n+1\), \(\lambda_ n(\omega)\), \(n=1,2,...\), on \((\Omega_ 1,F_ 1,P_ 1)\). Let \((\Omega^*,F^*,P^*)\) be the direct product of (\(\Omega\),F,P) and \((\Omega_ 1,F_ 1,P_ 1)\), and \(\tau_ n(\omega,\omega_ 1)\) be the time for passing through the points 1,2,...,n. The asymptotic behaviour of \(r_ n\) and \(x_ t\) as \(n\to \infty\) is given under the assumptions of 1) \(\lambda_ n=\eta_ n\), 2) \(\lambda_ n=n^{\alpha}\eta_ n\) and 3) \(\lambda_ n=n\eta_ n\), corresponding to the case of one-sided random walk, \(\phi\)-controllable process and pure birth process in a random medium, respectively, where \(\sum_{n}\lambda_ n^{-1}=\infty\) a.s. and \(\eta_ n's\) are i.i.d. on \(\Omega\). The basic results remain valid for more general supercritical branching processes and are useful in biology, genetics and chemistry.