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	~~{{Distinguish\|Feller-continuous process}}~~		I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. What me and my family members love is to climb but I'm considering on starting something new. My day job is a journey agent. Her family members life in Alaska but her husband wants them to move.<br><br>Visit my web site - free online tarot card readings ([http://www.onbizin.co.kr/xe/?document_srl=320614 visit the up coming site])

	~~{{Refimprove\|date=September 2009}}~~

	~~In [[probability theory]] relating~~ to ~~[[stochastic processes]]~~, a '~~''Feller process''' is a particular kind of [[Markov process]].~~

	~~==Definitions==~~

	~~Let ''X'' be a [[locally compact]] [[topological space]] with a [[countable set\|countable]] [[base (topology)\|base]]. Let ''C''<sub>0</sub>(''X'') denote the space of~~ all ~~real-valued [[continuous function]]s on ''X'' that [[vanish at infinity]], equipped with the [[sup norms\|sup-norm]] \|\|''f'' \|\|~~.

	~~A '''Feller semigroup''' on ''C''<sub>0</sub>(''X'')~~ is ~~a collection {''T''<sub>''t''</sub>}<sub>''t'' ≥ 0</sub> of positive [[linear map]]s from ''C''<sub>0</sub>(''X'')~~ to ~~itself such that~~
	* \|\|'~~'T''<sub>''t''</sub>''f'' \|\| ≤ \|\|''f'' \|\| for all ''t'' ≥ 0 and ''f'' in ''C''<sub>0</sub>(''X''), i.e~~.~~, it~~ is a ~~[[contraction mapping\|contraction]] (in the weak sense);~~
	* the [[semigroup]] property: ''T''<sub>''t'' + ''s''</sub> = ''T''<sub>''t''</sub> o''T''<sub>''s''</sub> for all ''s'', ''t'' ≥ 0;
	* lim<sub>''t'' → 0</sub>\|\|''T''<sub>''t''</sub>''f'' − ''f'' \|\| = 0 for every ''f'' in ''C''<sub>0</sub>(''X''). ~~Using the semigroup property, this is equivalent to the map ''T''<sub>''t''</sub>''f''  from ''t''~~ in ~~[0,∞)~~ to ~~''C''<sub>0</sub>(''X'') being [[right continuous]] for every ''f''~~.

	~~'''Warning''': This terminology is not uniform across the literature. In particular, the assumption that ''T''<sub>''t''~~<~~/sub~~> ~~maps ''C''~~<~~sub>0</sub~~>(~~''X'') into itself~~
	~~is replaced by some authors by the condition that it maps ''C''<sub>b</sub>(''X''), the space of bounded continuous functions, into itself.~~
	~~The reason for this is twofold: first, it allows to include processes that enter "from infinity" in finite time. Second, it is more suitable to the treatment of~~
	~~spaces that are not locally compact and for which the notion of "vanishing at infinity" makes no sense.~~

	~~A '''Feller transition function''' is a probability transition function associated with a Feller semigroup.~~

	~~A '''Feller process''' is a Markov process with a Feller transition function.~~

	~~== Generator ==~~

	~~Feller processes (or transition semigroups) can be described by their~~ [~~[infinitesimal generator]]. A function ''f'' in ''C''<sub>0</sub> is said to be in the domain of the generator if the uniform limit~~

	: ~~<math> Af = \lim_{t\rightarrow 0} \frac{T_tf - f}{t},</math>~~

	~~exists. The operator ''A'' is the generator of ''T<sub>t<~~/~~sub>'', and the space of functions on which it is defined is written as ''D<sub>A<~~/~~sub>''~~.

	~~A characterization of operators that can occur as the infinitesimal generator of Feller processes is given by the [[Hille-Yosida theorem]]~~. ~~This uses the resolvent of the Feller semigroup, defined below~~.

	~~== Resolvent ==~~

	~~The '''resolvent''' of a Feller process (or semigroup) is a collection of maps (''R<sub>λ</sub>'')<sub>''λ'' > 0</sub> from ''C''<sub>0</sub>(''X'') to itself defined by~~
	~~:<math>R_\lambda f = \int_0^\infty e^{-\lambda t}T_t f\,dt.</math>~~
	~~It can be shown that it satisfies the identity~~
	~~:<math>R_\lambda R_\mu = R_\mu R_\lambda = (R_\mu-R_\lambda)~~/~~(\lambda-\mu).</math>~~
	~~Furthermore, for any fixed ''λ'' > 0, the image of ''R<sub>λ</sub>'' is equal to the domain ''D<sub>A</sub>'' of the generator ''A'', and~~
	~~:<math>~~
	~~\begin{align}~~
	~~& R_\lambda = (\lambda - A)^{-1}, \\~~
	~~& A = \lambda - R_\lambda^{-1}.~~
	~~\end{align}~~
	</~~math>~~

	~~== Examples =~~=

	* Brownian motion and the ~~Poisson process are examples of Feller processes. More generally, every [[Lévy process]] is a Feller process.~~

	* [[Bessel process]]es are Feller processes.

	* Solutions to [[stochastic differential equation]]s with [[Lipschitz continuous]] coefficients are Feller processes.

	* Every Feller process satisfies the [[strong Markov property]]~~.<ref>Liggett, Thomas Milton ''Continuous-time Markov processes: an introduction'' (page 93, Theorem 3.3~~) ~~{{full\|date=November 2012}}</ref>~~

	~~== See also ==~~

	* [[Markov process]]
	* [[Markov chain]]
	* [[Hunt process]]
	* [[Infinitesimal generator (stochastic processes)]]

	~~==References==~~
	~~{{reflist}}~~

	~~{{Stochastic processes}}~~

	~~{{DEFAULTSORT:Feller Process}}~~
	~~[[Category:Stochastic processes]]~~

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clean up, removed orphan tag using AWB

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{{Distinguish|Feller-continuous process}}

{{Refimprove|date=September 2009}}

In [[probability theory]] relating to [[stochastic processes]], a '''Feller process''' is a particular kind of [[Markov process]].

==Definitions==

Let ''X'' be a [[locally compact]] [[topological space]] with a [[countable set|countable]] [[base (topology)|base]]. Let ''C''0(''X'') denote the space of all real-valued [[continuous function]]s on ''X'' that [[vanish at infinity]], equipped with the [[sup norms|sup-norm]] ||''f'' ||.

A '''Feller semigroup''' on ''C''0(''X'') is a collection {''T''''t''}''t'' ≥ 0 of positive [[linear map]]s from ''C''0(''X'') to itself such that
* ||''T''''t''''f'' || ≤ ||''f'' || for all ''t'' ≥ 0 and ''f'' in ''C''0(''X''), i.e., it is a [[contraction mapping|contraction]] (in the weak sense);
* the [[semigroup]] property: ''T''''t'' + ''s'' = ''T''''t'' o''T''''s'' for all ''s'', ''t'' ≥ 0;
* lim''t'' → 0||''T''''t''''f'' − ''f'' || = 0 for every ''f'' in ''C''0(''X''). Using the semigroup property, this is equivalent to the map ''T''''t''''f''  from ''t'' in [0,∞) to ''C''0(''X'') being [[right continuous]] for every ''f''.

'''Warning''': This terminology is not uniform across the literature. In particular, the assumption that ''T''''t'' maps ''C''0(''X'') into itself
is replaced by some authors by the condition that it maps ''C''b(''X''), the space of bounded continuous functions, into itself.
The reason for this is twofold: first, it allows to include processes that enter "from infinity" in finite time. Second, it is more suitable to the treatment of
spaces that are not locally compact and for which the notion of "vanishing at infinity" makes no sense.

A '''Feller transition function''' is a probability transition function associated with a Feller semigroup.

A '''Feller process''' is a Markov process with a Feller transition function.

== Generator ==

Feller processes (or transition semigroups) can be described by their [[infinitesimal generator]]. A function ''f'' in ''C''0 is said to be in the domain of the generator if the uniform limit

: <math> Af = \lim_{t\rightarrow 0} \frac{T_tf - f}{t},</math>

exists. The operator ''A'' is the generator of ''Tt'', and the space of functions on which it is defined is written as ''DA''.

A characterization of operators that can occur as the infinitesimal generator of Feller processes is given by the [[Hille-Yosida theorem]]. This uses the resolvent of the Feller semigroup, defined below.

== Resolvent ==

The '''resolvent''' of a Feller process (or semigroup) is a collection of maps (''Rλ'')''λ'' > 0 from ''C''0(''X'') to itself defined by
:<math>R_\lambda f = \int_0^\infty e^{-\lambda t}T_t f\,dt.</math>
It can be shown that it satisfies the identity
:<math>R_\lambda R_\mu = R_\mu R_\lambda = (R_\mu-R_\lambda)/(\lambda-\mu).</math>
Furthermore, for any fixed ''λ'' > 0, the image of ''Rλ'' is equal to the domain ''DA'' of the generator ''A'', and
:<math>
\begin{align}
& R_\lambda = (\lambda - A)^{-1}, \\
& A = \lambda - R_\lambda^{-1}.
\end{align}
</math>

== Examples ==

* Brownian motion and the Poisson process are examples of Feller processes. More generally, every [[Lévy process]] is a Feller process.

* [[Bessel process]]es are Feller processes.

* Solutions to [[stochastic differential equation]]s with [[Lipschitz continuous]] coefficients are Feller processes.

* Every Feller process satisfies the [[strong Markov property]].<ref>Liggett, Thomas Milton ''Continuous-time Markov processes: an introduction'' (page 93, Theorem 3.3) {{full|date=November 2012}}</ref>

== See also ==

* [[Markov process]]
* [[Markov chain]]
* [[Hunt process]]
* [[Infinitesimal generator (stochastic processes)]]

==References==
{{reflist}}

{{Stochastic processes}}

{{DEFAULTSORT:Feller Process}}
[[Category:Stochastic processes]]

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