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| {{Distinguish|Feller-continuous process}}
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| {{Refimprove|date=September 2009}}
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| In [[probability theory]] relating to [[stochastic processes]], a '''Feller process''' is a particular kind of [[Markov process]].
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| ==Definitions==
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| 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'' ||.
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| 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
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| * ||''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);
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| * the [[semigroup]] property: ''T''<sub>''t'' + ''s''</sub> = ''T''<sub>''t''</sub> o''T''<sub>''s''</sub> for all ''s'', ''t'' ≥ 0;
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| * 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''.
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| '''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
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| is replaced by some authors by the condition that it maps ''C''<sub>b</sub>(''X''), the space of bounded continuous functions, into itself.
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| 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
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| spaces that are not locally compact and for which the notion of "vanishing at infinity" makes no sense.
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| A '''Feller transition function''' is a probability transition function associated with a Feller semigroup.
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| A '''Feller process''' is a Markov process with a Feller transition function.
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| == Generator ==
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| 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
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| : <math> Af = \lim_{t\rightarrow 0} \frac{T_tf - f}{t},</math> | |
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| 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>''.
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| 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.
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| == Resolvent ==
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| 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
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| :<math>R_\lambda f = \int_0^\infty e^{-\lambda t}T_t f\,dt.</math>
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| It can be shown that it satisfies the identity
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| :<math>R_\lambda R_\mu = R_\mu R_\lambda = (R_\mu-R_\lambda)/(\lambda-\mu).</math>
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| 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
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| :<math>
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| \begin{align}
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| & R_\lambda = (\lambda - A)^{-1}, \\
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| & A = \lambda - R_\lambda^{-1}.
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| \end{align}
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| </math>
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| == Examples ==
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| * Brownian motion and the Poisson process are examples of Feller processes. More generally, every [[Lévy process]] is a Feller process.
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| * [[Bessel process]]es are Feller processes.
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| * Solutions to [[stochastic differential equation]]s with [[Lipschitz continuous]] coefficients are Feller processes.
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| * 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>
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| == See also ==
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| * [[Markov process]]
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| * [[Markov chain]]
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| * [[Hunt process]]
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| * [[Infinitesimal generator (stochastic processes)]]
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| ==References==
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| {{reflist}}
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| {{Stochastic processes}}
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| {{DEFAULTSORT:Feller Process}}
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| [[Category:Stochastic processes]]
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