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Thread: Markov processes and generator

  1. #1
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    Markov processes and generator

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

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

    The main difficulty comes from the diagonal elements of $\displaystyle P$.

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