# Math Help - Markov processes and generator

1. ## 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 $P$.

2. Writing the matrix $P(t)=exp(tQ)$ enables to get the result directly.