Does the domain of the generator of an Ito diffusion contain $C^2$ or just $C_c^2$?

Question

Does the domain of the generator of an Ito diffusion contain $C^2$ or just $C_c^2$?

From Wikipedia

(For the generator $A$) One can show that $C_c^2$, i.e. any compactly-supported $C^2$ (twice differentiable with continuous second derivative) function f, lies in $DA$ and that $$ Af(x) = \sum_{i} b_{i} (x) \frac{\partial f}{\partial x_i} (x) + \tfrac{1}{2} \sum_{i, j} \left( \sigma (x) \sigma (x)^{\top} \right)_{i, j} \frac{\partial^{2} f}{\partial x_i \, \partial x_{j}} (x), $$ This agrees with Oksendal's SDE book.

Also from Wikipedia:

the generator and characteristic operator agree for all $C^2$ functions $f$, in which case $$ \mathcal{A} f(x) = \sum_i b_i (x) \frac{\partial f}{\partial x_{i}} (x) + \tfrac1{2} \sum_{i, j} \left( \sigma (x) \sigma (x)^{\top} \right)_{i, j} \frac{\partial^{2} f}{\partial x_{i} \, \partial x_{j}} (x). $$

I was wondering if the second quote means the domain of the generator contains $C^2$?

Thanks and regards!

Answer 1

As far as I know, one can only prove (only meaning that that's the only proof I've seen) that, if a diffusion takes values in $E\subseteq \mathbb{R}^{n}$, and $\mathcal{A}$ is it's infinitesimal generator, than $C^{2}_{c}(E)\subset \mathcal{D}(\mathcal{A})$, where $C^{2}_{c}(E)$ is the space of $C^{2}$ functions with compact support, taking values on $E$.

Hope that helps.