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

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

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

$${P}_{t+s}(x,A)=\int {P}_{s}(x,dy){P}_{t}(y,A).$$ 
