A note about Khoshnevisan--Xiao conjecture

Abstract

Khoshnevisan and Xiao showed in [Ann. Probab. 33 (2005) 841--878] 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.

Published at http://dx.doi.org/10.1214/009117906000000197 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Levy processes: Capacity and Hausdorff dimension