# Probability transitions for branching process

by kai_sikorski
Tags: branching, probability, process, transitions
 PF Gold P: 162 Working through a paper that uses this result about branching processes. Can't seem to figure out a way to connect the dots. Anyone have any suggestions? Let Zt be a branching process where each individual gives birth to a single offspring at rate β > 1 and dies at rate 1. The transition probabilities for Zt can be computed exactly and are given by $$P_{10}(t) = \frac{1 - e^{-(\beta-1)t}}{\beta - e^{-(\beta-1)t}}$$ $$P_{1k}(t) = (1 - P_{10}(t))(1 - \eta(t))\eta(t)^{k-1}$$ $$\eta(t) = \frac{1 - e^{-(\beta-1)t}}{1 - \frac{1}{\beta}e^{-(\beta-1)t}}$$ I know that P1i satisfy the following system of equations $$P_i'(t) = Q P(t)$$ Where Q is the transition rate matrix (i think that's what it's called) and is given by $$Q = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & \\ 0 & -(1+\beta ) & 2 & \ddots & \vdots \\ 0 & \beta & -2(1+\beta ) & 3 & \\ \vdots & \ddots & 2\beta & -3(1+\beta)& \ddots \\ & \ldots & & \ddots & \ddots \\ \end{array} \right)$$ I have verified the above solution works, but I'm not sure how one would have gotten it. Any ideas?
 PF Gold P: 162 Oh I figured it out, mostly Let G be the generating function for Zt $G[s,t] = p_{10}(t) + p_{11}(t)s + p_{12}(t)s^2 + ...$ By using the differential equations for the ps, and manipulating some indices you can get a PDE for G $\partial_t G[s,t]-(1-s(1+\beta))+\beta s^2) \partial_s G[s,t] = 0$ $G[s,0] = s$ I can plug this into DSolve and it gives the right answer. However I'm having some trouble doing the method of characteristics on this. For the family of characteristics I get $s[t] = \frac{e^{t \beta +c}+e^{t+\beta c}}{e^{t \beta +c}+e^{t+\beta c} \beta }$ But I don't know how to pick c, so that I can set an initial condition on s[t]. Any ideas?