Feller process

In probability theory relating to , a Feller process is a particular kind of Markov process.

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

Warning: This terminology is not uniform across the literature. In particular, the assumption that T_t maps C₀(X) into itself is replaced by some authors by the condition that it maps C_b(X), the space of bounded continuous functions, into itself. The reason for this is twofold: first, it allows to include processes that enter "from infinity" in finite time. Second, it is more suitable to the treatment of spaces that are not locally compact and for which the notion of "vanishing at infinity" makes no sense.

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.

Feller processes (or transition semigroups) can be described by their . A function f in C₀ is said to be in the domain of the generator if the uniform limit

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.

A characterization of operators that can occur as the infinitesimal generator of Feller processes is given by the Hille-Yosida theorem. This uses the resolvent of the Feller semigroup, defined below.

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

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