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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!

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  • $\begingroup$ Your strategy does not seem to work: For the 1-D Brownian motion on $(0, \infty)$, the local time $L_t$ at zero is equal in distribution to $\max(0, -\inf_{s \in [0,t]} W_s)$ (by construction via Skorokhod's problem), where $W_s$ is the standard Brownian motion. But $\inf_{s \in [0,t]} W_s$ is equal in distribution to $W_0 - |W_s - W_0|$ (by reflection principle). In particular, $t \mapsto \mathbb{E}(L_t)$ is differentiable. $\endgroup$ – Mateusz Kwaśnicki Sep 27 '17 at 12:17
  • $\begingroup$ I just checked. Sorry I should've been more clear. It isn't differentiable when t=0 which is all I need. I'll adjust the question when I get home. $\endgroup$ – fast_and_fourier Sep 28 '17 at 2:15
  • $\begingroup$ [1/2] OK, if the process starts at the boundary, than $\mathbb{E}(L_t)$ indeed is expected to have infinite derivative at $t = 0$, but I do not have a proof. Regarding the other part of the question, I would try the usual reflection trick: the domain (and hence the process) can be mapped (locally) onto half-space, and one can try constructing a process on half-space through an appropriate process on entire space (with coefficients extended to full space by symmetry); just as the reflecting Brownian motion can be constructed from the usual Brownian motion by applying the absolute value... $\endgroup$ – Mateusz Kwaśnicki Sep 29 '17 at 8:33
  • $\begingroup$ [2/2] ...to the last coordinate. (Is this clearly written? If not, I can elaborate). However, although $a$ remains Lipschitz, $b$ becomes discontinuous, which might cause problems. Final note: Feller generators have rather wild domains; in particular, there are functions in the domain which are not $C^2$. $\endgroup$ – Mateusz Kwaśnicki Sep 29 '17 at 8:33

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