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- W1906275395 abstract "Let $(X,mathcal W)$ be a balayage space, $1in mathcal W$, 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 $u,vin mathcal W$ such that $u/vto 0$ at infinity. We suppose that there is a Green function $G>0$ for $X$, a metric $rho$ on $X$ and a decreasing function $gcolon[0,infty)to (0,infty]$ having the doubling property such that $Gapprox gcircrho$. 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 $B(z,r_z)$, $zin Z$, 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 $x_0in X$, the series $sum_{zin 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. Vondrav cek, and the author." @default.
- W1906275395 created "2016-06-24" @default.
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- W1906275395 date "2014-09-26" @default.
- W1906275395 modified "2023-09-27" @default.
- W1906275395 title "Unavoidable sets and Wiener's test for Hunt processes" @default.
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