Infinitesimal generator of a Markov process

Remarks

Example: for a continuous-time Markov chain on a finite set $S$ with rate matrix $Q$, one has $T_t = e^{tQ}$, so $(T_t f - f)/t \to Qf$ for every $f$, and $A = Q$ with $D(A)$ the whole space. Example: for Brownian motion on $\R^d$, every twice continuously differentiable $f$ with $f$, its first and its second derivatives in $C_0(\R^d)$ lies in $D(A)$, and $Af = \tfrac12 \Delta f$. Non-example: for Brownian motion on $\R$, the function $f(x) = e^{-x^2} \abs{x}$ is in $C_0(\R)$ but not in $D(A)$, since it is not differentiable at $0$. For $f \in D(A)$, Dynkin's formula says that $f(X_t) - f(X_0) - \int_0^t Af(X_s)\, ds$ is a martingale. The semigroup satisfies $\frac{d}{dt} T_t f = A T_t f = T_t A f$ for $f \in D(A)$, the backward and forward equations.