a Feller semigroup
/ FEL-er /
A Feller semigroup is the operator-theoretic object that sits behind a Feller process: it is the specific kind of strongly continuous contraction semigroup that arises as the transition semigroup of a nice Markov process, and that one can recognize purely from properties of the operators, before any process is constructed. It is the bridge on which Hille-Yosida is specialized from abstract Banach-space semigroups to Markov dynamics.
Definition: on the Banach space C_0(E) (continuous functions vanishing at infinity on a locally compact separable metric space, with sup-norm), a Feller semigroup is a family (P_t)_(t >= 0) of operators that is (1) a C_0 (strongly continuous) semigroup, (2) sub-Markov positive — 0 <= f <= 1 implies 0 <= P_t f <= 1 (and conservative/Markov if additionally P_t 1 = 1) — and (3) a contraction, which the positivity and P_t 1 <= 1 already deliver. The characterization theorem is a Markov version of Hille-Yosida: a densely-defined operator A generates a Feller semigroup if and only if A is closable, its closure satisfies the range condition (lambda I - A) has dense range for some/all lambda > 0, and A satisfies the POSITIVE MAXIMUM PRINCIPLE — if f in D(A) attains a non-negative maximum at x_0, then (A f)(x_0) <= 0. The positive maximum principle is the operator fingerprint of probability-conservation: it is what forces A to have the integro-differential Lévy-type form (drift + non-negative-definite diffusion + a non-negative jump kernel), via the Courrège theorem.
Why it matters: the Feller-semigroup framework is how one constructs and classifies the well-behaved Markov processes from their generators, and the positive maximum principle is the precise extra condition (beyond Hille-Yosida) that distinguishes generators of MARKOV semigroups from generators of arbitrary contraction semigroups. A caveat in honesty: 'Feller semigroup' is sometimes defined as sub-Markov (allowing mass loss, P_t 1 <= 1, modelling killing/explosion) and sometimes as conservative (P_t 1 = 1); the distinction matters because a sub-Markov semigroup corresponds to a process that may be killed or explode, and conservativeness is a real, checkable property, not a given.
The positive maximum principle pins down generator form: for A f = (1/2) f'' + b f', if f has a non-negative max at x_0 then f'(x_0) = 0 and f''(x_0) <= 0, so (A f)(x_0) <= 0 automatically. By Courrège's theorem, ANY operator obeying the principle has the Lévy-type form drift + diffusion + jump kernel — there are no other Feller generators.
The positive maximum principle is the operator fingerprint of probability conservation and forces the Lévy-type generator form.
Watch the convention: 'Feller semigroup' may mean sub-Markov (P_t 1 <= 1, allowing killing/explosion) or conservative (P_t 1 = 1); conservativeness is a property to verify, not assume.