# Feller process

Let $E$ be a LCCB space (locally compact with a countable base; usually a subset of $\mathbb{R}^{n}$ for some $n\in\mathbb{N}$) and $C_{0}(E)=C_{0}(E,\mathbb{R})$ be the space of continuous functions  on $E$ that vanish at infinity. (We may write $C_{0}$ as shorthand.) A Feller semigroup on $C_{0}(E)$ is a family of positive  linear operators $T_{t},t\geq 0$, on $C_{0}(E)$ such that

• $T_{0}=Id$ and $||T_{t}||\leq 1$ for every $t\in T$, i.e. $\{T_{t}\}_{t\in T}$ is a family of contracting maps;

• $T_{t+s}=T_{t}\circ T_{s}$ (the semigroup property);

• $\lim_{t\downarrow 0}||T_{t}f-f||=0$ for every $f\in C_{0}(E)$.

A probability transition function associated with a Feller semigroup is called a Feller transition function. A Markov process having a Feller transition function is called a Feller process.

## References

Title Feller process FellerProcess 2013-03-22 16:12:40 2013-03-22 16:12:40 mcarlisle (7591) mcarlisle (7591) 6 mcarlisle (7591) Definition msc 60J35 Feller semigroup Feller transition function Feller process LCCB