Feller process
Appearance
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 {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.
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.