Feller semigroups and strongly continuous semigroups are the same thing?

Question

Revuz and Yor in their book "Continuous Martingales and Brownian Motion" use the following definition of a Feller semigroup: A Feller semigroup on $C_0(S)$ (continuous functions vanishing at infinity) is a family $T_t,\, t\geq 0$, of positive linear operators on $C_0(S)$ such that

• $T_0=Id$ and $||T_t||\leq 1$ for every $t$;
• $T_{t+s}=T_t \circ T_s$ for any pair $s, t\geq 0$;
• $\lim_{t\downarrow 0} ||T_t f - f||=0$ for every $f\in C_0(S)$

However, in more functional-analytic sources I see that if a family of bounded linear operators on some Banach space satisfies the above properties, then it is called a strongly continuous or a $C_0$ semigroup.

Questions:

• What is the actual definition of a function being strongly continuous? I cannot find it anywhere.
• Is it correct to assume that the two names are interchangeable?
• Am I right that probabilists prefer to use "Feller semigroups" because they give rise to Feller processes, while functional analysts just use the other name?

Thanks a lot.

Answer 1

What is the actual definition of a function being strongly continuous? I cannot find it anywhere.

A semigroup on $C_0(S)$ is strongly continuous iff $$\lim_{t \to 0} \|T_t f-f\| = 0 \quad \text{for all $f \in C_0(S)$}.$$

Is it correct to assume that the two names are interchangeable?

Yes. A family of operators $(T_t)_{t \geq 0}$ is called

• a semigroup if $T_{t+s} = T_t \circ T_s$ and $T_0 = \text{id}$.

• bounded if there exists $M>0$ such that $\|T_t\| \leq M$ for all $t \geq 0$

This means that a family of operators $T_t: C_0(S) \to C_0(S)$, $t \geq 0$, is a Feller semigroup if and only if it is a bounded (with $M=1$) strongly continuous semigroup. Note that Revuz' definition contains, implicitly, that $T_t$ maps $C_0(S)$ to $C_0(S)$, i.e. $T_t f \in C_0(S)$ for any $f \in C_0(S)$; this property is called Feller property.

Am I right that probabilists prefer to use "Feller semigroups" because they give rise to Feller processes, while functional analysts just use the other name?

Yes, there is a one-to-one relation between Feller semigroups and Feller processes, and therefore probabilists prefer to use this name.

Note, however, that there are different notions of "Feller semigroups" in the literature; therefore it is sometimes clearer to use the term $C_0$-semigroup to avoid any confusion.

Answer 2

It should be noted that there is one difference between a Feller semigroup and a strongly continuous contraction (i.e. bounded with $M=1$) semigroup on $C_0(S)$: A Feller semigroup is always positive, i.e. every positive function is mapped to positive one, while a strong continuous contraction semigroup is not necessarily positive. To be rigorous, a Feller semigroup is a strongly continuous, positive, contraction semigroup on $C_0(S)$.