Liouville Property, Wiener's Test and Unavoidable Sets for Hunt Processes

Hansen W (2016)
Potential Analysis 44(4): 639-653.

Zeitschriftenaufsatz | Veröffentlicht| Englisch
 
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Abstract / Bemerkung
Let be a balayage space, , or - equivalently - let be the set of excessive functions of a Hunt process on a locally compact space X with countable base such that separates points, every function in is the supremum of its continuous minorants and there exist strictly positive continuous such that u/v -> 0 at infinity. We suppose that there is a Green function G > 0 for X, a metric rho for X and a decreasing function having the doubling property such that G approximate to g omicron rho. Assuming that the constant function 1 is harmonic and balls of (X, rho) are relatively compact, it is shown that every positive harmonic function on X 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 a 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) a X, the series 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. Vondraek, and the author.
Stichworte
Hunt process; Balayage space; Unavoidable set; Zero-one law; Green; function; Equilibrium measure; Capacity; Doubling property; Liouville; property; Wiener's criterion; Levy process
Erscheinungsjahr
2016
Zeitschriftentitel
Potential Analysis
Band
44
Ausgabe
4
Seite(n)
639-653
ISSN
0926-2601
eISSN
1572-929X
Page URI
https://pub.uni-bielefeld.de/record/2904098

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Hansen W. Liouville Property, Wiener's Test and Unavoidable Sets for Hunt Processes. Potential Analysis. 2016;44(4):639-653.
Hansen, W. (2016). Liouville Property, Wiener's Test and Unavoidable Sets for Hunt Processes. Potential Analysis, 44(4), 639-653. doi:10.1007/s11118-015-9523-0
Hansen, W. (2016). Liouville Property, Wiener's Test and Unavoidable Sets for Hunt Processes. Potential Analysis 44, 639-653.
Hansen, W., 2016. Liouville Property, Wiener's Test and Unavoidable Sets for Hunt Processes. Potential Analysis, 44(4), p 639-653.
W. Hansen, “Liouville Property, Wiener's Test and Unavoidable Sets for Hunt Processes”, Potential Analysis, vol. 44, 2016, pp. 639-653.
Hansen, W.: Liouville Property, Wiener's Test and Unavoidable Sets for Hunt Processes. Potential Analysis. 44, 639-653 (2016).
Hansen, Wolfhard. “Liouville Property, Wiener's Test and Unavoidable Sets for Hunt Processes”. Potential Analysis 44.4 (2016): 639-653.