Feller process: Difference between revisions
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==Definitions== |
==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]]. |
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 |
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''), |
* ||''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 |
* 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''). |
* 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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A '''Feller transition function''' is a probability transition function associated with a Feller semigroup. |
A '''Feller transition function''' is a probability transition function associated with a Feller semigroup. |
Revision as of 05:06, 8 May 2011
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 base. Let C0(X) denote the space of all real-valued continuous functions on X that vanish at infinity, equipped with the sup-norm ||f ||.
A Feller semigroup on C0(X) is a collection {Tt}t ≥ 0 of positive linear maps from C0(X) to itself such that
- ||Ttf || ≤ ||f || for all t ≥ 0 and f in C0(X), i.e., it is a contraction (in the weak sense);
- the semigroup property: Tt + s = Tt oTs for all s, t ≥ 0;
- limt → 0||Ttf − f || = 0 for every f in C0(X). Using the semigroup property, this is equivalent to the map Ttf from t in [0,∞) to C0(X) being right continuous for every f.
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 C0 is said to be in the domain of the generator if the uniform limit
exists. The operator A is the generator of Tt, and the space of functions on which it is defined is written as DA.
Resolvent
The resolvent of a Feller process (or semigroup) is a collection of maps (Rλ)λ > 0 from C0(X) to itself defined by
It can be shown that it satisfies the identity
Furthermore, for any fixed λ > 0, the image of Rλ is equal to the domain DA of the generator A, and
Examples
- Brownian motion and the Poisson process are examples of Feller processes. More generally, every Lévy process is a Feller process.
- Bessel processes are Feller processes.
- Solutions to stochastic differential equations with Lipschitz continuous coefficients are Feller processes.
See also