PlanetMath: Feller process

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Feller process

(Definition)

Let 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 that vanish at infinity. (We may write as shorthand.) A Feller semigroup on is a family of positive linear operators $T_t, t \geq 0$ , on such that

and $\vert\vert T_t\vert\vert \leq 1$ 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);
$\lim_{t \downarrow 0} \vert\vert T_tf - f\vert\vert = 0$ 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.

"Feller process" is owned by mcarlisle.

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Also defines:

Feller semigroup, Feller transition function, Feller process, LCCB

Keywords:

Feller Markov random stochastic process semigroup

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Cross-references: probability transition function, property, semigroup, maps, linear operators, positive, vanish at infinity, continuous functions, subset, base, countable, locally compact
There is 1 reference to this entry.

This is version 3 of Feller process, born on 2006-09-01, modified 2006-09-01.
Object id is 8307, canonical name is FellerProcess.
Accessed 2776 times total.

Classification:

AMS MSC:

60J35 (Probability theory and stochastic processes :: Markov processes :: Transition functions, generators and resolvents)

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