I am currently investigating the domain of the infinitesimal generator of a reflected stochstic differential equation (for a smooth and bounded domain) with Lipshitz coefficients. Namely SDEs of the form

\begin{equation} dX_t=a(X_t,t)dt+b(X_t,t)dW_t+n(X_t)dL_t \end{equation}

where $n$ is the unit normal derivative on the boundary and $L$ is the assosciated local time process.

I want to show that the functions in the domain of this infinitesimal generator must have zero Neumann Boundary conditions. By doing a quick computation using Ito's Lemma it seems that this can be proven if it is shown that \begin{equation} \mathbb{E}(L_{t}) \end{equation} is not differentiable with respect to $t$.

So my question is: Is there a reference which explicitly shows that the domain of the infinitesimal generator are $C^2$ functions with zero Neumann Boundary conditions?

A potential proof strategy would also be much appreciated!

isdifferentiable. $\endgroup$ – Mateusz Kwaśnicki Sep 27 '17 at 12:17