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Math Help - Markov processes and generator

  1. #1
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    Markov processes and generator

    Let P(t) be a differentiable semi-goup of Markov transition matrices with the infinitesimal matrix (generator) Q. Assuming that Q_{ij}\neq 0 \ for \ 1\leq i,j\leq r, how do you prove that for every t>0 all the matrix entries of P(t) are >0?
    (Prove then that there is a unique stationary distribution \pi for the semi-group of transition matrices.)

    Note there is a hint: represent Q as (Q+cI)-cI with a constant c sufficiently large so that to make all the elements of the matrix Q+cI non-negative.

    The main difficulty comes from the diagonal elements of P.

    Thanks for your help.
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  2. #2
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    Writing the matrix P(t)=exp(tQ) enables to get the result directly.
    End of thread.
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