Feller process

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

${\displaystyle Af=\underset{t\to 0}{lim}\frac{T_{t}f-f}{t},}$

exists. The operator A is the generator of T_t, and the space of functions on which it is defined is 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.

Resolvent

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

${\displaystyle R_{\lambda }f={\int }_0^{\mathrm{\infty }}{e}^{-\lambda t}{T}_{t}fdt.}$

It can be shown that it satisfies the identity

${\displaystyle R_{\lambda }{R}_{\mu }=R_{\mu }{R}_{\lambda }=(R_{\mu }-R_{\lambda })/(\lambda -\mu ).}$

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

${\displaystyle \begin{array}{cc}& R_{\lambda }=(\lambda -A{)}^{-1},\\ & A=\lambda -R_{\lambda }^{-1}.\end{array}}$

Examples

• 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.
• Every adapted right continuous Feller process on a filtered probability space ${\displaystyle (\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t{)}_{t\ge 0})}$ satisfies the strong Markov property with respect to the filtration ${\displaystyle ({\mathcal{F}}_{t^{+}}{)}_{t\ge 0}}$, i.e., for each ${\displaystyle ({\mathcal{F}}_{t^{+}}{)}_{t\ge 0}}$-stopping time ${\displaystyle \tau }$, conditioned on the event ${\displaystyle \{\tau <\mathrm{\infty }\}}$, we have that for each ${\displaystyle t\ge 0}$, ${\displaystyle X_{\tau +t}}$ is independent of ${\displaystyle {\mathcal{F}}_{{\tau }^{+}}}$ given ${\displaystyle X_{\tau }}$.