probability transition function

A probability transition function (p.t.f., or just t.f. in context) on a measurable space $(\mathrm{\Omega },ℱ)$ is a family $P_{s,t}$, of transition probabilities on $(\mathrm{\Omega },ℱ)$ such that for every three real numbers , the family the Chapman-Kolmogorov equation

$$\int P_{s,t}(x,dy)P_{t,v}(y,A)=P_{s,v}(x,A)$$

for every $x\in \mathrm{\Omega }$ and $A\in ℱ$. The t.f. is said to be if $P_{s,t}$ depends on $s$ and $t$ only through their $t-s$. In this case, we write $P_{t,0}=P_t$ and the family $\{P_t,t\ge 0\}$ is a semigroup, and the Chapman-Kolmogorov equation reads

$$P_{t+s}(x,A)=\int P_s(x,dy)P_t(y,A).$$