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.