I consider a diffusion process $$ dX_t = b(X_t) dt + \sigma(X_t) dB_t. $$
From the general theory, we know that if $f\in C^2(\mathbb{R})$ and has a compact support, then the infinitesimal generator is \begin{equation} Af = \frac{\sigma^2(x)}{2}\frac{d^2 f(x)}{dx^2} + b(x)\frac{df(x)}{dx}. \end{equation}
In the special case, when $b = 0$ and $\sigma = 1$ we have the infinitesimal generator of Brownian motion of the form $$ Af = \frac{1}{2}\frac{d^2 f(x)}{dx^2}. $$ This formula also holds true even if $f\in C^2(\mathbb{R})$ (without the assumption about compact support).
My question is:
Can we also exclude the assumption about the compactness of support for the infinitesimal generator of diffusion process in one dimension? Or maybe can we state another weaker condition than the general one for $f$ function?
Any references will be very useful for me.