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.

单词	FellerProcess
释义	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.
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