regular harmonic function is harmonic. Moreover, the Green function for the set $D$ (if it exists) is harmonic in $x$ on $D \setminus \{y\}$.

Let us note two simple observations.

\[prop:2\] For any $x,y \in {\mathbb{R}}$ we have $$\Go(x,y) {\leqslant}H(x) \wedge H(y).$$

For any $\lambda>0$ we have $$\begin{aligned}
    \Go^{\lambda}(x,y) &= u^{\lambda}(y-x)-{\mathbb{E}}^x e^{-\lambda T_0} u^{\lambda}(y-T_0) \\ &=u^{\lambda}(y-x) - u^{\lambda}(y)h^{\lambda}(x) \\ &= -K^{\lambda}(y-x) + K^{\lambda}(-x) + K^{\lambda}(y) - \frac{K^{\lambda}(-x)K^{\lambda}(y)}{u^{\lambda}(0)}.
    \end{aligned}$$ Recall that $K^{\lambda} {\geqslant}0$. By the proof of [@GR17 Proposition 2.2] we get that $K^{\lambda}$ is subadditive on ${\mathbb{R}}$. Thus, $K^{\lambda}(y) {\leqslant}K^{\lambda}(x) + K^{\lambda}(y-x)$ and consequently, $$\Go^{\lambda}(x,y) {\leqslant}-K^{\lambda}(y-x)+K^{\lambda}(-x)+K^{\lambda}(y) {\leqslant}K^{\lambda}(x)+K^{\lambda}(-x).$$

\[prop:3\] For any $|x| \in (0,R)$ we have $${\mathbb{E}}^x \big[ {\tau_{(-R,R)}}\wedge T_0 \big] {\leqslant}2RH(x).$$

By Proposition \[prop:2\], $${\mathbb{E}}^x \big[ {\tau_{(-R,R)}}\wedge T_0 \big] = \int_{-R}^R G_{(-R,0)\cup (0,R)}(x,y)\,dy {\leqslant}\int_{-R}^R \Go(x,y)\, dy {\leqslant}2RH(x).$$

Fluctuation theory {#subsec:fluctuation}
------------------

Before embarking on further results we need some introduction from the fluctuation theory of Lévy processes. The general reference here is the book of Bertoin [@Bertoin96].

First, let us observe that the condition implies that $\mathbf{X}$ is of unbounded variation. Indeed, suppose the converse; then $\mathbf{X}$ can be written as a difference of two subordinators. It follows that ${\operatorname{Re}}\psi$ has at most linear growth and hence, $$\int_0^{\infty} \frac{d\xi}{1+{\operatorname{Re}}\psi(\xi)}=\infty,$$ which 