approximation of a Feller semi-group with the infinitesimal generator

Question

Let $T_t$ a Feller semigroup (see this) and let $(A,D(A))$ its infinitesimal generator.

If A is a bounded operator it is easy to show that the Feller semi-group is $e^{tA}$.

Is this formula always true if $(A,D(A))$ is a diffusion on a compact manifold? I know that in some Sobolev space the operator has to be bounded, but i'm not sure if this implies that the operator is strongly bounded.

In the general case in which the operator is unbounded is it still possible to approximate the semigroup as

$$ T_t f=f+ tAf +\frac{t^2}{2}A^2f+ \cdots +\frac{t^{n}}{n!}A^n f + o(t^n) $$

If we restrict the domain of the semi-group to the smooth functions? Such formula is widely used in the context of numerical approximation of stochastic processes, but i can't find any reference in which is proven.

Answer 1

You look for the concept of analytic (or entire) vector. If you have a strongly continuous group, then they are dense, otherwise it might happen that there is only the 0 vector (for nilpotent semigroups). See e.g.

Engel, Klaus-Jochen; Nagel, Rainer, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). ZBL0952.47036.

Exercise II.3.12 (2).