a Feller process
/ FEL-er /
A Feller process is the well-behaved class of continuous-time Markov process for which the abstract operator theory and the concrete sample-path theory meet cleanly. It is defined by a regularity condition on its transition semigroup — the Feller property — that is just strong enough to guarantee good paths (càdlàg, right-continuous with left limits) and the strong Markov property, while being weak enough to cover Brownian motion, Lévy processes, and a vast range of diffusions and jump processes.
Definition via the semigroup: a Markov process is Feller if its transition operators P_t act on C_0(E) (continuous functions vanishing at infinity on a locally compact, separable state space) so that (i) P_t maps C_0(E) into C_0(E) for each t (smoothing / the spatial Feller property), and (ii) the semigroup is strongly continuous, ||P_t f - f||_sup -> 0 as t -> 0 for every f in C_0(E) (the temporal Feller property). A semigroup with these properties is a Feller semigroup, and the central structural theorems then follow: a Feller process has a càdlàg modification (so we may and do work with right-continuous paths with left limits), and it satisfies the STRONG Markov property — the random-time restart that an arbitrary Markov process can lack. The generator A is then a closed, densely-defined operator on C_0(E) characterized by Hille-Yosida together with the positive maximum principle.
Why it matters: Feller is the standard hypothesis under which the whole toolkit — Dynkin's formula, hitting-time analysis, weak-convergence/martingale-problem constructions, potential theory — is licensed, because the analytic property (acting nicely on C_0) buys the probabilistic properties (good paths, strong Markov) for free. Honest caveats: the Feller property is genuinely a hypothesis, not automatic; spatial Feller (P_t : C_0 -> C_0) and temporal strong continuity are distinct conditions both needed; and many natural Markov processes (e.g. some with killing, or on non-locally-compact spaces) are NOT Feller, requiring the weaker C_b-Feller or right-process frameworks instead.
Every Lévy process is Feller: its semigroup acts by convolution, (P_t f)(x) = E[ f(x + X_t) ], which maps C_0(R^d) to C_0(R^d) and is strongly continuous since X_t -> 0 in probability as t -> 0. Hence Lévy processes have càdlàg paths and are strong Markov — the foundation for studying their jumps and first passages.
The Feller property of Lévy semigroups is what guarantees their càdlàg paths and strong Markovianity.
Feller is not free: spatial (P_t : C_0 -> C_0) and temporal strong continuity are separate hypotheses, and processes with killing or on non-locally-compact spaces often fail it.