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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 $L$.

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  • $\begingroup$ 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. $\endgroup$ – saz Jul 8 at 5:25

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