# Domain of the Generator of a Bessel process

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.

• In dimension 1, generators of diffusion processes were already discussed by Feller in the fifites. The domain is described in terms of the scale function (and the speed 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. – Mateusz Kwaśnicki Mar 10 at 8:13
• @MateuszKwaśnicki This is interesting. I had a look at the paper that you linked but it seems to apply to matrix semigroups as opposed to semigroups arising from SDEs. Does Feller have other resources for the continuous time case? – fast_and_fourier Mar 10 at 11:49
• Yes, I must've messed up the link, sorry. I'll fix it later today. – Mateusz Kwaśnicki Mar 10 at 13:32
• I am writing this in a hurry, so please excuse me if the references are still not correct. I think I meant a series of three papers by Feller in the Annals: 1, 2 and 3. A standard reference now is Itô–McKean, I think. I can look it up and provide a more detailed reference, if you like. – Mateusz Kwaśnicki Mar 10 at 21:01
• @MateuszKwaśnicki yes a detailed reference certainly would help a lot. – fast_and_fourier Mar 12 at 6:51