Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$ \begin{align} \rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t} \end{align} where $(W_{t})_{t\geq 0}$ is a Brownian motion. We take the usual probability space and filtration.

It is well known that the Bessel Process in this case is reflecting. However I see many papers, eg this one, that say that the infinitesimal generator, if we take the Banach Space of continuous and bounded functions with the usual norm, is the following:

\begin{align} \mathcal{L}f=\frac{1}{2}\frac{d^2f}{dr^2}+\frac{1+2\nu}{2}\frac{1}{r}\frac{df}{dr} \end{align}

with the domain \begin{align} \mathcal{D}(\mathcal{L})=\{f\in C_{b}(\mathbb{R_{+}})\,|\,\mathcal{L}f\in C_{b}(\mathbb{R_{+}}), \lim_{r\rightarrow 0}r^{1-2\nu}f'(r)=0\}. \end{align} The Bessel Process is somewhat difficult to manage, since in this case it isn't always a semi-martingale, however even in the case where it is a semi-martingale I cannot quite see where the boundary condition comes in $\lim_{r\rightarrow 0}r^{1-2\nu}f'(r)=0$. I suspect my attempts with Ito's Lemma are missing some critical ideas.

I would like to know if there is a reference that goes through the proof of this result.

scale function(and thespeed measure). Roughly speaking, with reflection at the boundary, one has to assume that the derivative of the composition of $f$ with the scale function (which happens to be $r^{2 - 2\nu}$ in our case) has to be zero. $\endgroup$ – Mateusz Kwaśnicki Mar 10 at 8:13