a strongly continuous contraction semigroup
A strongly continuous contraction semigroup — a C_0 contraction semigroup — is the precise functional-analytic object that the Hille-Yosida theory characterizes, and it is the abstract home of a Markov transition semigroup. It strips the picture down to its essential axioms on a Banach space, forgetting the probabilistic origin, so that one can ask which operators generate such a flow and what regularity P_t f inherits.
Let (X, ||.||) be a Banach space. A family (P_t)_(t >= 0) of bounded linear operators on X is a C_0 contraction semigroup if: (1) P_0 = I; (2) the semigroup law P_(t+s) = P_t P_s holds for all t, s >= 0; (3) strong continuity, meaning for every f in X the orbit t -> P_t f is continuous from [0, infinity) into X, equivalently ||P_t f - f|| -> 0 as t -> 0+; and (4) contraction, ||P_t|| <= 1 for all t (each P_t is non-expanding). Note 'strong' continuity is continuity of the orbits, NOT continuity of t -> P_t in operator norm (uniform continuity) — uniform continuity is much stronger and forces a bounded generator, which excludes essentially all interesting differential generators. Strong continuity is exactly the weak regularity that still permits unbounded generators like the Laplacian.
Why it matters: this is the class on which P_t = e^(tA) makes rigorous sense for an unbounded A, the resolvent is a Laplace transform of the semigroup, and Hille-Yosida tells you when a given A actually generates one. For Markov processes the contraction is automatic (averaging shrinks the sup-norm) and the semigroup is additionally positive with P_t 1 = 1; strong continuity on C_0(E) is precisely the Feller condition. An honest caveat: many natural Markov semigroups are NOT strongly continuous on the space of all bounded measurable functions (only on C_0 or L^p), which is why the correct Banach space must be fixed at the outset.
The heat semigroup (P_t f)(x) = E[ f(x + B_t) ] on C_0(R) is C_0: ||P_t f - f||_sup -> 0 as t -> 0 by uniform continuity of f, and ||P_t f||_sup <= ||f||_sup (contraction). But it is NOT uniformly continuous in t, consistent with its generator being the unbounded operator A = (1/2) d^2/dx^2.
Strong (orbit-wise) continuity, not uniform continuity, is what lets the Laplacian be a legitimate generator.
Strong continuity is continuity of orbits t -> P_t f, not of t -> P_t in operator norm; demanding the latter would force a bounded generator and rule out diffusions.