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 $$ \lim_{ (x,y) \to (x_0,z_0)} \dfrac{G^{\Omega}_X(x,y)}{G^{\Omega}_X(x_0,y)},\quad x_0 \in \Omega, z_0 \in \partial \Omega$$ 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 $$ M^{\Omega}(x,z_0):=\lim_{ y \to z_0} \dfrac{G^{\Omega}_Y(x,y)}{G^{\Omega}_Y(x_0,y)}, \quad z_0 \in \partial \Omega$$ exists, which is again the Martin kernel of $Y^{\Omega}$.

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

Thank you very much.

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.

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 $$ \lim_{ (x,y) \to (x_0,z_0)} \dfrac{G^{\Omega}_X(x,y)}{G^{\Omega}_X(x_0,y)},\quad x_0 \in \Omega, z_0 \in \partial \Omega$$ 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 fixed $x_0 \in \Omega$ and $x \in \Omega$, the limit $$ M^{\Omega}(x,z_0):=\lim_{ y \to z_0} \dfrac{G^{\Omega}_Y(x,y)}{G^{\Omega}_Y(x_0,y)}, \quad z_0 \in \partial \Omega$$ exists, which is again the Martin kernel of $Y^{\Omega}$.

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

Thank you very much.

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 $$ \lim_{ (x,y) \to (x_0,z_0)} \dfrac{G^{\Omega}_X(x,y)}{G^{\Omega}_X(x_0,y)},\quad x_0 \in \Omega, z_0 \in \partial \Omega$$ 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 $$ M^{\Omega}(x,z_0):=\lim_{ y \to z_0} \dfrac{G^{\Omega}_Y(x,y)}{G^{\Omega}_Y(x_0,y)}, \quad z_0 \in \partial \Omega$$ exists, which is again the Martin kernel of $Y^{\Omega}$.

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

Thank you very much.