# Build a Markov process from a transition semigroup

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

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

• math.stackexchange.com/questions/1351521/… It doesn't have an answer, but I think my comment should finish the job. I'm just concerned about issues with the domain of $P_t$.
– Ian
Commented Jul 13, 2015 at 10:26

Yes, one can go the way back if there are assumptions on the state space $(E,\mathcal{E})$.
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.
• With "Borel space", do you mean that $\mathcal E$ should be the Borel $\sigma$ algebra of $E$? So, if the state space is $\mathbb R^n$, for example, $\mathcal E$ should be the Borel sets of $\mathbb R^n$? Commented Jul 13, 2015 at 13:06
• No, I give a link to the encyclopedia of math. Any Polish space is a Borel space, so $\mathbb{R}^n$ with the Borel sets (which is induced by some complete and separable metric) is also a Borel space. Commented Jul 13, 2015 at 14:16
• Define $p(t,x,A) := T_t \mathbf{1}_A (x)$ for $t\geq 0$, $x \in E$ and $A \in \mathcal{E}$; $\mathbf{1}_A$ denotes the indicator function of $A$; this will lead to a probability kernel if we know a bit more on $T_t$. To answer your comment precisely: could you modify the question such that it includes a definition or a reference for "transition semigroup" or "Markov semigroup"? Often nonegative conservative semigroups of linear operators on Banach spaces are considered and $B_b(E)$ is in most cases not such a Banach space. Commented Jul 14, 2015 at 6:39