Necessary and Sufficient Conditions for the Continuity of Local Time of Levy Processes

Let u₁(x) be the 1-potential kernel density for a Levy process, let φ²(x) = 2u₁(0) - u₁(x) - u₁(-x), let φ̄ be the monotone rearrangement of φ and let I(φ̄) = ∫₀₊ φ(u)u⁻¹(log(1/u))-1/2 du. Barlow and Hawkes proved that if I(φ̄) < ∞, then the local time has a jointly continuous version. In this paper it is shown that if I(φ̄) < ∞, then the local time is not continuous.