Feller process

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

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[edit] 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 which vanish at infinity.

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),
  • the semigroup property: Tt + s = Tt oTs for all s, t ≥ 0,
  • limt → 0||Ttf - f || = 0 for every f in C0(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.

[edit] 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

Af = \lim_{t\rightarrow 0} \frac{T_tf - f}{t},

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

[edit] Resolvent

The resolvent of a Feller process (or semigroup) is a collection of maps (Rλ)λ > 0 from C0(X) to itself defined by

R_\lambda f = \int_0^\infty e^{-t}P_t f\,dt.

It can be shown that it satisfies the identity

RλRμ = RμRλ = (Rμ − Rλ) / (λ − μ).

Furthermore, for any fixed λ > 0, the image of Rλ is equal to the domain DA of the generator A, and

\begin{align} & R_\lambda = (\lambda - A)^{-1}, \\ & A = \lambda - R_\lambda^{-1}. \end{align}

[edit] Examples

  • Brownian motion and the Poisson process are examples of Feller processes. More generally, every Levy process is a Feller process.

[edit] See also

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