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), i.e., it is a contraction (in the weak sense);
the semigroup property: Tt + s = Tt ∘Ts for all s, t ≥ 0;
limt → 0||Ttf − f || = 0 for every f in C0(X). Using the semigroup property, this is equivalent to the map Ttf from t in [0,∞) to C0(X) being right continuous for every f.
Warning: This terminology is not uniform across the literature. In particular, the assumption that Tt maps C0(X) into itself
is replaced by some authors by the condition that it maps Cb(X), the space of bounded continuous functions, into itself.
The reason for this is twofold: first, it allows including 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.
Generatoredit
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
exists. The operator A is the generator of Tt, and the space of functions on which it is defined is written as DA.
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.
Resolventedit
The resolvent of a Feller process (or semigroup) is a collection of maps (Rλ)λ > 0 from C0(X) to itself defined by
It can be shown that it satisfies the identity
Furthermore, for any fixed λ > 0, the image of Rλ is equal to the domain DA of the generator A, and
Every adapted right continuous Feller process on a filtered probability space satisfies the strong Markov property with respect to the filtration , i.e., for each -stopping time, conditioned on the event , we have that for each , is independent of given .[1]
^Rogers, L.C.G. and Williams, DavidDiffusions, Markov Processes and Martingales volume One: Foundations, second edition, John Wiley and Sons Ltd, 1979. (page 247, Theorem 8.3)