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Probability transitions for branching process

  1. Feb 27, 2012 #1

    kai_sikorski

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    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

    [tex]P_{10}(t) = \frac{1 - e^{-(\beta-1)t}}{\beta - e^{-(\beta-1)t}}[/tex]
    [tex]P_{1k}(t) = (1 - P_{10}(t))(1 - \eta(t))\eta(t)^{k-1}[/tex]
    [tex]\eta(t) = \frac{1 - e^{-(\beta-1)t}}{1 - \frac{1}{\beta}e^{-(\beta-1)t}}[/tex]

    I know that P1i satisfy the following system of equations

    [tex] P_i'(t) = Q P(t) [/tex]

    Where Q is the transition rate matrix (i think that's what it's called) and is given by

    [tex] 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) [/tex]

    I have verified the above solution works, but I'm not sure how one would have gotten it. Any ideas?
     
    Last edited: Feb 27, 2012
  2. jcsd
  3. Feb 27, 2012 #2

    kai_sikorski

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    Gold Member

    Oh sorry, the IC is pi(0) =δi1
     
  4. Feb 28, 2012 #3

    kai_sikorski

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    Gold Member

    Oh I figured it out, mostly

    Let G be the generating function for Zt

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

    By using the differential equations for the ps, and manipulating some indices you can get a PDE for G

    [itex] \partial_t G[s,t]-(1-s(1+\beta))+\beta s^2) \partial_s G[s,t] = 0[/itex]
    [itex] G[s,0] = s [/itex]

    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

    [itex] s[t] = \frac{e^{t \beta +c}+e^{t+\beta c}}{e^{t \beta +c}+e^{t+\beta c} \beta }[/itex]

    But I don't know how to pick c, so that I can set an initial condition on s[t]. Any ideas?
     
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