Let $B_t$ be the standard Brownain Motion on $\mathbb R$. It is known that its generator is $$Lf=\frac{1}{2}f''$$ and it is defined on function $f\in C^2(\mathbb R, \mathbb R)$.
It is possible to define the generator $L$ on a bigger space, for istance for $f$ in the Sobolev space $H^2(\mathbb R)$?
Thank you very much