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);
• $\lim_{t \downarrow 0} ||T_tf - f|| = 0$ 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.

Bibliography

1

D. Revuz & M. Yor, Continuous Martingales and Brownian Motion, Third Edition Corrected. Volume 293, Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 2005.