  Let $(X,\mathcal W)$ be a balayage space, $1\in \mathcal W$, or -
equivalently - let $\mathcal W$ be the set of excessive functions of a Hunt
process on a locally compact space $X$ with countable base such that $\mathcal
W$ separates points, every function in $\mathcal W$ is the supremum of its
continuous minorants and there exist strictly positive continuous $u,v\in
\mathcal W$ such that $u/v\to 0$ at infinity. We suppose that there is a Green
function $G>0$ for $X$, a metric $\rho$ on $X$ and a decreasing function
$g\colon[0,\infty)\to (0,\infty]$ having the doubling property such that
$G\approx g\circ\rho$.
  Assuming that the constant function $1$ is harmonic and balls are relatively
compact, is is shown that every positive harmonic function is constant
(Liouville property) and that Wiener's test at infinity shows, if a given set
$A$ in $X$ is unavoidable, that is, if the process hits $A$ with probability
one, wherever it starts.
  An application yields that locally finite unions of pairwise disjoint balls
$B(z,r_z)$, $z\in Z$, which have a certain separation property with respect to
a suitable measure $\lambda$ on $X$ are unavoidable if and only if, for
some/any point $x_0\in X$, the series $\sum_{z\in Z} g(\rho(x_0,z))/g(r_z) $
diverges.
  The results generalize and, exploiting a zero-one law for hitting
probabilities, simplify recent work by S. Gardiner and M. Ghergu, A. Mimica and
Z. Vondra\v cek, and the author.
