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Given a markov model with transition rate matrix $Q$, where the probability of each state at time $t$ is given by $P(t) = P(0)e^{Qt}$, where e is the matrix exponential, the steady state probabilities should be given by taking the $\lim_{t\to\infty}P(t)$.

Is this the best way, or are there alternatives, and if so, how do you solve them with those methods? If not, how does one take the limit of the matrix exponential as t goes to infinity so that you can arrive at an analytical solution.

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Find the left eigenvector to eigenvalue 0 of $Q$ and there is your stationary $P_\infty$. Or in formulas $$P_\infty Q = 0.$$

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  • $\begingroup$ I thought $P_\infty Q = P_\infty$, not zero. $\endgroup$
    – bmillare
    Apr 7 '11 at 20:13
  • $\begingroup$ Nevermind, I'm thinking $P_\infty e^{Q}$ $\endgroup$
    – bmillare
    Apr 7 '11 at 20:14
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    $\begingroup$ @bmillare: From $P_\infty Q =0$, it follows that $P_{\infty} e^{Q t} = P_{\infty}$ --- exactly what you want for a stationary solution. $\endgroup$
    – Fabian
    Apr 7 '11 at 20:15

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