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

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	2013-03-22 16:12:40
Last modified on	2013-03-22 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