Feller process

In mathematics, a Feller process is a particular kind of Markov process.

Definitions

Let "X" be a locally compact topological space with a countable base. Let "C"₀("X") denote the space of all real-valued continuous functions on "X" which vanish at infinity.

A Feller semigroup on "C"₀("X") is a collection {"T"_"t"}_{"t" ≥ 0} of positive linear maps from "C"₀("X") to itself such that
* ||"T"_"t""f" || ≤ ||"f" || for all "t" ≥ 0 and "f" in "C"₀("X"),
* 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").

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:$Af\; =\; lim\_\{t\; ightarrow\; 0\}\; frac\{T\_tf\; -\; f\}\{t\}$,exists. The operator "A" is the generator of "T_t", and the space of functions on which it is defined is wriiten as "D_A".

Resolvent

The resolvent of a Feller process (or semigroup) is a collection of maps ("R_λ")_{"λ" > 0} from "C"₀("X") to itself defined by:$R\_lambda\; f\; =\; int\_0^infty\; e^\{-lambda\; t\}P\_t\; f,dt.$It can be shown that it satisfies the identity:$R\_lambda\; R\_mu\; =\; R\_mu\; R\_lambda\; =\; (R\_mu-R\_lambda)/(lambda-mu).$Furthermore, for any fixed "λ > 0", the image of "R_λ" is equal to the domain "D_A" 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

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