# probability transition function

A probability transition function (p.t.f., or just t.f. in context) on a measurable space $(\Omega,\mathcal{F})$ is a family $P_{s,t}$, $0\leq s of transition probabilities on $(\Omega,\mathcal{F})$ such that for every three real numbers $s, the family the Chapman-Kolmogorov equation

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

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

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

## References

Title probability transition function ProbabilityTransitionFunction 2013-03-22 16:12:37 2013-03-22 16:12:37 mcarlisle (7591) mcarlisle (7591) 8 mcarlisle (7591) Definition msc 60J35 probability transition function homogeneous probability transition function Chapman-Kolmogorov equation