Feller process

Let X be a locally compact Hausdorff 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 ||. From analysis, we know that C₀(X) with the sup norm is a Banach space.

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 $\backslash circ$T_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.

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 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.

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

: $Af\; =\; \backslash lim\_\{t\backslash rightarrow\; 0\}\; \backslash frac\{T\_tf\; -\; f\}\{t\},$

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.

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

:$R\_\backslash lambda\; f\; =\; \backslash int\_0^\backslash infty\; e^\{-\backslash lambda\; t\}T\_t\; f\backslash ,dt.$

It can be shown that it satisfies the identity

:$R\_\backslash lambda\; R\_\backslash mu\; =\; R\_\backslash mu\; R\_\backslash lambda\; =\; (R\_\backslash mu-R\_\backslash lambda)/(\backslash lambda-\backslash mu).$

Furthermore, for any fixed λ > 0, the image of R_λ is equal to the domain D_A of the generator A, and

:$$

\begin{align}

& R_\lambda = (\lambda - A)^{-1}, \\

& A = \lambda - R_\lambda^{-1}.

\end{align}

• 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.{{citation needed|date=December 2015}}
• Every adapted right continuous Feller process on a filtered probability space $(\backslash Omega,\; \backslash mathcal\{F\},\; (\backslash mathcal\{F\}\_t)\_\{t\backslash geq\; 0\})$ satisfies the strong Markov property with respect to the filtration $(\backslash mathcal\{F\}\_\{t^+\})\_\{t\backslash geq\; 0\}$, i.e., for each $(\backslash mathcal\{F\}\_\{t^+\})\_\{t\backslash geq\; 0\}$-stopping time $\backslash tau$, conditioned on the event , we have that for each $t\backslash ge\; 0$, $X\_\{\backslash tau\; +\; t\}$ is independent of $\backslash mathcal\{F\}\_\{\backslash tau^+\}$ given $X\_\backslash tau$.Rogers, L.C.G. and Williams, David Diffusions, Markov Processes and Martingales volume One: Foundations, second edition, John Wiley and Sons Ltd, 1979. (page 247, Theorem 8.3)

• Markov process
• Markov chain
• Hunt process
• Infinitesimal generator (stochastic processes)

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Category:Markov processes