Build a Markov process from a transition semigroup

Question

Suppose that we have a Markov process $X$ with time-homogeneus transition probability function given by $p(t,x,A)$, with $t>0, x\in E, A \in \mathcal E$, where $(E,\mathcal E)$ is the state space.

Associated with $p$, and thus with $X,$ we have the transition semigroup $P_t$ of $X$ which is defined as: $$P_tf(x):=\int_E f(y)p(t,x,dy)=\mathbb E (f(X^x_t))$$ for each $f\in B_b(E)$, where $B_b(E)$ denotes the bounded Borel measurable functions on $E$.

Also, we know that, given a probability transition function on $E$, there exists a unique Markov process having that transition probability function.

Can we do something similar with the transition semigroup? I mean, given a Markov semigroup $T_t$ on $E$, does there exist a Markov process $X$ with transition probability function $p$ associated to $T_t$ through the formula

$$T_tf(x)=\int_E f(y)p(t,x,dy)?$$

Answer 1

Yes, one can go the way back if there are assumptions on the state space $(E,\mathcal{E})$.

Given a transition semigroup you can define a transition probability function using indicator functions.

And from the transition probability function you can construct a Markov process using the Kolmogorov extension theorem.

Well, this works if $(E,\mathcal{E})$ has some additional properties. In Kallenberg's Foundations of Modern Probability (2003), Theorem 6.16, it is required that $(E,\mathcal{E})$ is a Borel space. Other sources use other conditions, see e.g. Bogachev, Measure Theory I, Chapter 7.7. If one restricts the time set to $T= \mathbb{N}_0$ instead of $T = [0,\infty)$, then one can use a theorem of Ionescu Tulcea which does not require anything on the state space.