Feller Processes and Semigroups

For convenience, we can have a look at the list of materials contained in this lecture first. Note here we always consider the time-homogenous Markov processes. • Transition kernels μt, t ≥ 0 ←→ transition operators semigroup Tt, t ≥ 0. • Feller semigroup Tt, t ≥ 0 • Feller processes • existence • every Feller process has a cadlag version • regularization theorem for submartingale • strong Markov property • Blumental 0-1 law • Lévy processes • Generator of Feller semigroup • existence • Dynkin’s formula • extended generator for Markov process • generator of Lévy process • three basic building blocks • generator of Feller process • application (heat equation) Now we start from the definition of the transition operator Tt. For any Markov transition kernels μt(·, ·) on (S,S), f : S → R bounded or nonnegative, define transition operator Tt: