I'm reading a paper by Villemonais and Mèlèard , where it is used the Perron Frobenius theorem for a continuous in time Mkv Chain. It is stated that there exists a positive eigenvalue for the matrix $ P_{t_0}$ (where $ (P_{t})$ is a sub-Markovian semi-group of probabilities) with $t_0$ such that $P_{t_0}$ has only positive entries.
But I had read Darroch and Seneta (1967) where the theorem is stated for $Q$ the infinitesimal generator.
What is the relationship between both versions? Are both correct?
Where can I find the theorem for continuous time properly stated (and proved if possible)?
How does this statement change when considering a Mkv Chain on an infinite state space?
The classical version of Perron Frobenius theorem is stated for positive matrices (as in the MV paper you're citing). See for instance the wikipedia entry https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem#Positive_matrices. There exist more powerful versions of the Perron-Frobenius theorem, but this one should cover your need.
I don't find any reference to an infinitesimal generator in Darroch and Seneta 1965. However, in general, if you have a Perron-Frobenius theorem for an infinitesimal generator $Q$, then you immediately obtain the same kind of conclusion for $P_{t_0}$ through the equality $P_{t_0}=e^{Qt_0}$.