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 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 {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),
• 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).

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 wriiten as D_A.

Examples

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