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Feller process
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(Definition)
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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);
- $\lim_{t \downarrow 0} ||T_tf - f|| = 0$ 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.
- 1
- D. Revuz & M. Yor, Continuous Martingales and Brownian Motion, Third Edition Corrected. Volume 293, Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 2005.
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"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 3929 times total.
Classification:
| AMS MSC: | 60J35 (Probability theory and stochastic processes :: Markov processes :: Transition functions, generators and resolvents) |
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Pending Errata and Addenda
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