Unavoidable collections of balls for processes with isotropic unimodal Green function

Let us suppose that we have a right continuous Markov semigroup on $R^d$, $d\ge 1$, such that its potential kernel is given by convolution with a function $G_0=g(|\cdot|)$, 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 $\alpha$-stable semigroups (Riesz potentials), but also more general isotropic L\'evy 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 $R^d$ 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 $B(z,r_z)$, $z\in Z$, which is unavoidable, $\sum_{z\in Z} g(|z|)/g(r_z)=\infty$. The converse is proven assuming, in addition, that, for some $\varepsilon>0$, $|z-z'|\ge \varepsilon |z| (g(|z|)/g(r_z))^{1/d}$, whenever $z,z'\in Z$, $z\ne z'$. 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 $r_z=g(\phi(|z|))$, where $\phi$ is a decreasing function. The results generalize and, exploiting a zero-one law, simplify recent work by A. Mimica and Z. Vondracek.