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⊂ℝ₊ under a symmetric Lévy process X in 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 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.