Feller Semigroup

A Semigroup satisfying the properties:

• $\Vert T_{t}f\Vert \le \Vert f\Vert \forall t\ge 0,f\in C_0(X)$, i.e. $T_t$ is a contraction.
• $T_{t+s}=T_t\circ T_s\forall t,s\ge 0$, i.e. it is a Semigroup
• $\underset{t\to 0}{\lim }\Vert T_{t}f-f\Vert =0$, i.e. $T$ is right-continuous.

In the above,

• $X$ is a locally compact hausdorff space with a countable base
• $C_0(X)$ is the space of real-valued continuous functions on $X$ that vanish at infinity