I'm reading about Support of Potential measures for Lévy Processes. $\Sigma$ denotes the support of $U(0,\cdot),$ where $U(x,B)=\int_{0}^{\infty}P_{x}(X_{t}\in B)\,\mathrm dt$ is the potential measure of the process $(X_{t})_{t\geq 0},$ and qualify the points in $\Sigma$ as possible.
Then the book says:
- $x\in\mathbb{R}^{d}$ is possible if and only if $\forall\epsilon>0,\exists t>0$ such that $P(X_{t}\in B(x,\epsilon))>0$ if and only if $P(T_{B}<\infty)>0$ for every ball $B$ centred at $x.$ Here $T_{B}=\inf\{t>0:X_{t}\in B\}.$
- If $x,y\in\Sigma$ then $x+y\in\Sigma.$
I'm trying to prove 1) and 2) but I'm stuck. For 1) I have the second equivalence because of the definition of $T_{B},$ but the other is not clear to me. I think I'm not understanding the meaning of $x\in\Sigma.$
For 2) the book says that consider $x,y\in\Sigma$ and $B(z,\eta)$ where $\eta>0.$ Is enough to apply Markov property at the first passage time into $B(x,\epsilon)$ to see that $P(T_{B(x+y,2\epsilon)}<\infty)>0.$ How the Markov property works here? I don't understand how the author utilizing here.
Any kind of help is thanked in advanced.