Feller process

Let $E$ be a LCCB space (locally compact with a countable base; usually a subset of ${ℝ}^n$ for some $n\in \mathbb{N}$) and $C_0(E)=C_0(E,ℝ)$ 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 $T_t,t\ge 0$, on $C_0(E)$ such that

• $T_0=Id$ and $||T_t||\le 1$ 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.