Let $(X_t)$ an diffusion Itô process, i.e. a solution of $$dX_t=b(X_t)dt+\sigma (X_t)dB_t.$$ The infinitesimal generator of $(X_t)$ is $$Af(x)=\lim_{t\to 0^+}\frac{\mathbb E^x[f(X_t)]-f(x)}{t},$$ where $\mathbb E^x$ is the expectation wrt $\mathbb P^x$.
Q1) What represent exactly $Af(x)$ for $X_t$ ? For example, for a Brownian motion, if $f$ is $C^2$ then $$A f(x)=\frac{1}{2}\Delta f(x).$$
But I don't really understand which information does $A$ give is. Is it a sort of derivative of $X_t$ ?
Q2) What is exactely the measure $\mathbb P^x$ ? I know it is $\mathbb P^x\{X_t\in A\}=\mathbb P(\{X_t\in A\}\mid \{X_0=x\}),$ But does it mean that on $(\Omega ,\mathcal F,\mathbb P^x)$ we have that $\mathbb P\{X_0=x\}=1$ ? (i.e. is deterministic).
Q1) It seems that this link could help.
Q2) Also see this link for a formal definition of conditional probability. Even, the existence of regular conditional probabilities is a non-trivial fact even for simple cases like $\mathbb{R}^2$ equipped with Borel sigma algebra.