Probability transitions for branching process

[kai_sikorski]

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 Z_t 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 Z_t 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 P_1i 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?

 

[kai_sikorski]

Oh sorry, the IC is p_i(0) =δ_i1

 

[kai_sikorski]

Oh I figured it out, mostly

Let G be the generating function for Z_t

$G[s,t] = p_{10}(t) + p_{11}(t)s + p_{12}(t)s^2 + ...$

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?