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A note about Khoshnevisan-Xiao conjecture. (English) Zbl 1101.60030

Summary: D. Khoshnevisan and Y. Xiao [Ann. Probab. 33, No. 3, 841–878 (2005; Zbl 1072.60040)] showed that the statement about almost surely vanishing Bessel-Riesz capacity of the image of a Borel set \(G \subset\mathbb{R}_+\) under a symmetric Lévy process \(X\) in \(\mathbb{R}^d\) is equivalent to the vanishing of a deterministic \(f\)-capacity for a particular function \(f\) defined in terms of the characteristic exponent of \(X\). The authors conjectured that a similar statement is true for all Lévy processes in \(\mathbb{R}^d\). We show that the conjecture is true provided we extend the definition of \(f\) and require certain integrability conditions which cannot be avoided in general.

MSC:

60G51 Processes with independent increments; Lévy processes
60J45 Probabilistic potential theory
60J25 Continuous-time Markov processes on general state spaces

Citations:

Zbl 1072.60040
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References:

[1] Khoshnevisan, D. and Xiao, Y. (2005). Lévy processes: Capacity and Hausdorff dimension. Ann. Probab. 33 841–878. · Zbl 1072.60040 · doi:10.1214/009117904000001026
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