I am reading a book on diffusion process and I noticed that there is a concept about semi group approach and I feel unclear about this: what is the domain of the generator $A$? Or how to describe it in a more elementary way?
The infinitesimal generator $A$ is defined as $$A\phi:=\lim_{t\rightarrow 0} \frac{T_t\phi-\phi}{t}$$ for any bounded continuous function $f$ and $t>0$, $$T_tf(x)=\int_{\mathbb{R}^n}f(y)P_t(x,dy),\quad x\in\mathbb{R}^n$$
However, when I try to search about it on google, all I see is abstract algebra which seems to be irrelavent to my question. I appreciate if someone would share some relavent material here.
Addition: Someone told me that $\left(\overline{\left\{\phi: \phi \in C_{2} \text { with compact support }\right\}},\|\cdot\|_{D(A)}\right)$ with $\|\phi\|_{D(A)}:=\|\phi\|_{C_{0}}+\|A \phi\|_{C_{0}}$ is the domain but unsure of it. $C_2$ is the set of twice continuously differentiable function and $C_0$ is the set of continuous function which is vanishing at infinity.
Does anyone have any idea about it?