Feller process

Not to be confused with Feller-continuous process.

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 {T_t}_t ≥ 0 of positive linear maps from C₀(X) to itself such that

• ||T_tf || ≤ ||f || for all t ≥ 0 and f in C₀(X), i.e., it is a contraction (in the weak sense);
• the semigroup property: T_t + s = T_t oT_s for all s, t ≥ 0;
• lim_t → 0||T_tf − f || = 0 for every f in C₀(X). Using the semigroup property, this is equivalent to the map T_tf  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

$Af=\lim _{t\rightarrow 0}{\frac {T_{t}f-f}{t}},$

exists. The operator A is the generator of T_t, and the space of functions on which it is defined is written as D_A.

Resolvent

The resolvent of a Feller process (or semigroup) is a collection of maps (R_λ)_λ > 0 from C₀(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

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

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)