Suppose $X_t^x$ is the solution to $$ d X^x_t=b(X_t)dt+dL_t, $$ where $L$ is a rotational symmetric $\alpha-$stable process with $\alpha\in (0,1]$, $b$ is Lipchitz.

Assume $\Gamma\subseteq \mathbb{R}^d$ and $\ dim(\Gamma)<d-1$. Is it possible to prove $\Gamma$ is a polar set, which means $$ {\bf P}_x(T_\Gamma<\infty)=0, \quad \forall x\in \mathbb{R}^d? $$ where $T_{\Gamma}:=\inf\{t>0: X_t\in\Gamma \}$.

Suppose $X_t$ is the solution to $$ d X_t=b(X_t)dt+dL_t,\quad X_0=x. $$ where $L$ is a rotational symmetric $\alpha-$stable process with $\alpha\in (0,1]$, $b$ is Lipchitz.

Assume $\Gamma\subseteq \mathbb{R}^d$ and $\ dim(\Gamma)<d-1$. Is it possible to prove $\Gamma$ is a polar set, which means $$ {\bf P}_x(T_\Gamma<\infty)=0, \quad \forall x\in \mathbb{R}^d? $$ where $T_{\Gamma}:=\inf\{t>0: X_t\in\Gamma \}$.

Suppose $X_t$ is a Feller process and $q(x,\xi)$ is the symbol of $X$, which satisfies $$ \lim_{t\downarrow 0} {\bf E}_x \frac{f(X_t)-f(X_0)}{t}=-(2\pi)^{-d/2} \int_{\mathbb{R}^d} e^{ix\cdot\xi}q(x,\xi)\hat{f}(\xi)\ d \xi, $$ for all $f\in C^\infty_c(\mathbb{R}^d)$.

Assume $$ \sup_{x\in \mathbb{R}^d}|q(x, \xi)|\leq C(1+|\xi|), \quad\forall\xi\in \mathbb{R}^d $$ and $\Gamma\subseteq \mathbb{R}^d$ with $\ dim(\Gamma)<d-1$. Is it possible to prove $\Gamma$ is a polar set, which means $$ {\bf P}_x(T_\Gamma<\infty)=0, \quad \forall x\in \mathbb{R}^d? $$ Here $T_{\Gamma}:=\inf\{t>0: X_t\in\Gamma \}$.

Remark: If $X_t$ is a Levy process, I know how to prove this, but if $X_t$ is a general Felller process, I have no idea. One example of $q(x,\xi)$ is $$ q(x,\xi)= |\xi|^{\alpha}+i b(x)\cdot\xi. $$

Suppose $X_t^x$ is the solution to $$ d X^x_t=b(X_t)dt+dL_t, $$ where $L$ is a rotational symmetric $\alpha-$stable process with $\alpha\in (0,1]$, $b$ is Lipchitz.

Assume $\Gamma\subseteq \mathbb{R}^d$ and $\ dim(\Gamma)<d-1$. Is it possible to prove $\Gamma$ is a polar set, which means $$ {\bf P}_x(T_\Gamma<\infty)=0, \quad \forall x\in \mathbb{R}^d? $$ where $T_{\Gamma}:=\inf\{t>0: X_t\in\Gamma \}$.