# Generator Brownian Motion

I'm referring to the construnction of the generator of the Brownian Motion. I marked the points I'm referring in the following here.

1. Why do we need to consider $$C_0^2$$?
2. Where did we show that $$B$$ is a $$C_0$$-process?

I only see that for $$f \in C_0^2 \subseteq C_0 \Rightarrow P_tf \in C_0^2$$ but there could be $$f \in C_0$$ such that $$P_tf \notin C_0$$, couldn't?

($$C_0$$-processes are defined as the Markov processes which generate a $$C_0$$-Semigroup (Def))

3. Using the Taylor-formula at the green marked lines it seems that $$B^i_h=x^i$$. Why is this the case?

4. Why is $$C_0^2 \subseteq D(A)$$ ?

$$D(A) := \left\{ f\in L : \operatorname*{\mathit{s}-lim}_{h\downarrow 0} \frac 1 h [P_h f-f] \text{ exists in L}\right\}$$ for a semigroup $$P_tf$$ of linear contractions on a Banach space $$L$$.

• Please avoid using images and links - take your time to type all the information which is needed for your question. This highly increases the likelihood that somebody will read (and answer) your question. – saz Jul 8 at 5:25