Transition function is a Markov semigroup?

Question

How does the transition function in a Markov process become a Markov semigroup in time homogeneous Markov processes?

Thanks a lot.

Answer 1

Let's call your state space $(X, \mathcal{B})$, your process $X_t$, and your transition function $p : [0,\infty) \times X \times \mathcal{B} \to [0,1]$, so that $p(t,x,B) = P_x(X_t \in B)$. For bounded measurable $f : X \to \mathbb{R}$, define $T_t f(x) = E_x[f(X_t)] = \int_X f(y) p(t,x,dy)$. It is clear that $T_t$ is Markovian; i.e. if $0 \le f \le 1$ then $0 \le T_t f \le 1$. And it follows from the Markov property of $X_t$ that $T_t$ is a semigroup, i.e. $T_t T_s f = T_{t+s} f$.