Definition of Martin kernels

Question

Let $\Omega \subset \mathbb{R}^n$ $(n \ge 3)$ be a bounded $C^{1,1}$ domain and let $X$ be a Markov process in $\Omega$. My question is regarding the existence of the Green function and Martin kernel of $X$.

Let $X$ be a symmetric $2\alpha$-stable process and $X^\Omega$ the subprocess of $X$ killed upon leaving $\Omega$. It is well-known that the infnitesimal-generator of $X^{\Omega}$ is the (restricted) fractional Laplacian $(-\Delta)^{\alpha}$. The existence of Green function $G^{\Omega}$ and Martin kernel $M^{\Omega}$ of $X$ are established in Estimates on Green functions and Poisson kernels for symmetric stable processes https://link.springer.com/article/10.1007/s002080050232. Furthermore, the joint limit $$ \lim_{ (x,y) \to (x_0,z_0)} \frac{G^\Omega_X(x,y)}{G^\Omega_X(x_0,y)},\quad x_0 \in \Omega, z_0 \in \partial \Omega$$ exists.

Now we consider a censored $2\alpha$-stable process $Y^{\Omega}$ whose infinitesimal-generator is the regional fractional Laplacian $(-\Delta_\Omega)^\alpha$. It is also known that for every reference point $x_0 \in \Omega$ and $x \in \Omega$, the limit $$ M^\Omega(x,z_0):=\lim_{ y \to z_0} \frac{G^\Omega_Y(x,y)}{G^\Omega_Y(x_0,y)}, \quad z_0 \in \partial \Omega$$ exists, which is again the Martin kernel of $Y^\Omega$.

My question is whether the joint limit $$ \lim_{ (x,y) \to (x_0,z_0)} \frac{G^\Omega_Y(x,y)}{G^\Omega_Y(x_0,y)},\quad x_0 \in \Omega, z_0 \in \partial \Omega$$ exists, like in the case of $X$.

Thank you very much.

Answer 1

I do know a reference for this particular result — although I am completely sure it is true!

The original Bogdan–Burdzy–Chen paper does not even provide a complete proof of the existence of the boundary limit. Instead, it refers to the case of usual fractional Laplacian. The proof of joint convergence is also very similar to the corresponding one for the usual fractional Laplacian, but I understand this is not sufficient for your needs.

Technically, you can refer to my paper with Tomasz Juszczyszyn, DOI:10.1007/s11118-017-9616-z. Remarks 3 and 4 there seem to address your question in a much greater generality. This is still somewhat awkward (one reuses the boundary result for an interior point to get equicontinuity of harmonic functions), but at least it does not seem to involve any hand-waving.

I guess one can find an even better reference in one of the papers by Panki Kim, Renming Song and Zoran Vondraček, but unfortunately I do not have enough time to search it now.