I'm studying the transition semigroup associated to a Markov Process, in particular the Hille-Yosida theorem and the Martingale Problem. In my notes I found : "If $\{T_t\}_t$ is a strongly continuous semigroup then there exist a unique Markov process with that given transition semigroup". Why?
In general, if I have a transition semigroup, how can I construct the Markov process associated?