Let \(\phi\) be a complete Bernstein function satisfying
(H1): there exist constants \(0<\delta_1\leq\delta_2<1\) and \(a_1,a_2>0\) such that \(a_1\lambda^{\delta_1}\phi(t) \leq \phi(\lambda t) \leq a_2\lambda^{\delta_2}\phi(t)\) for \(\lambda,t \geq 1\);
(H2): there exist constants \(0<\delta_3\leq\delta_4<1\) and \(a_3,a_4>0\) such that \(a_3\lambda^{\delta_4}\phi(t) \leq \phi(\lambda t) \leq a_4\lambda^{\delta_3}\phi(t)\) for \(\lambda,t \leq 1\).
Let \(X\) be a rotationally invariant Lévy process in \(\mathbb{R}^d\), \(d> 2(\delta_2 \vee \delta_4)\), with characteristic exponent \(\phi(|\xi|^2)\). The authors prove:
For every \(a > 0\), there exists a constant \(C = C(a,\phi)>1\) such that, for any \(r \geq 1\), any open set \(U \subset \bar{B}(0, r)^c\) and any nonnegative function \(u\) on \(\mathbb{R}^d\) which is regular harmonic with respect to \(X\) in \(U\) and vanishes a.e. on \(\bar{B}(0, r)^c \setminus U\), it holds that \[ C^{-1} K_U(x,0) \int_{B(0,ar)} u(z)\,dz \leq u(x) \leq C K_U(x,0) \int_{B(0,ar)} u(z)\,dz, \] for \(x\in U\cap \bar{B}(0,ar)^c\), where \(K_U\) denotes the Poisson kernel of \(X\) in \(U\times \bar{U}^c\).
In particular, the boundary Harnack principle at infinity holds.
The Martin boundary at infinity with respect to \(X\) of any open set \(D\) which is \(\kappa\)-fat at infinity consists of exactly one point \(\partial_\infty\). This point is a minimal Martin boundary point.
These results may be applied for a large class of subordinate Brownian motions.