Unavoidable collections of balls for processes with isotropic unimodal Green function

Let us suppose that we have a right continuous Markov semigroup on Rd, d ≥ 1, such that its potential kernel is given by convolution with a function G0 = g(| · |), where g is decreasing, has a mild lower decay property at zero, and a very weak decay property at infinity. This captures not only the Brownian semigroup (classical potential theory) and isotropic α-stable semigroups (Riesz potentials), but also more general isotropic Lévy processes, where the characteristic function has a certain lower scaling property, and various geometric stable processes. There always exists a corresponding Hunt process. A subset A of Rd is called unavoidable, if the process hits A with probability 1, wherever it starts. It is known that, for any locally finite union of pairwise disjoint balls B(z, rz), z ∈ Z, which is unavoidable, ∑ z∈Z g(|z|)/g(rz) =∞. The converse is proven assuming, in addition, that, for some ε > 0, |z − z′| ≥ ε|z|(g(|z|)/g(rz)), whenever z, z′ ∈ Z, z 6= z′. It also holds, if the balls are regularly located, that is, if their centers keep some minimal mutual distance, each ball of a certain size intersects Z, and rz = g(φ(|z|)), where φ is a decreasing function. The results generalize and, exploiting a zero-one law, simplify recent work by A. Mimica and Z. Vondraček.