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 ${T}_{t},t\ge 0$, on ${C}_{0}(E)$ such that

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

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${T}_{t+s}={T}_{t}\circ {T}_{s}$ (the semigroup property);

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${lim}_{t\downarrow 0}{T}_{t}ff=0$ 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
 1 D. Revuz & M. Yor, Continuous Martingales^{} and Brownian Motion^{}, Third Edition Corrected. Volume 293, Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 2005.
Title  Feller process 

Canonical name  FellerProcess 
Date of creation  20130322 16:12:40 
Last modified on  20130322 16:12:40 
Owner  mcarlisle (7591) 
Last modified by  mcarlisle (7591) 
Numerical id  6 
Author  mcarlisle (7591) 
Entry type  Definition 
Classification  msc 60J35 
Defines  Feller semigroup 
Defines  Feller transition function 
Defines  Feller process 
Defines  LCCB 