Necessary and sufficient conditions for joint continuity of Feller Brownian local times

Determine necessary and sufficient conditions on the Lévy measure η governing a Feller Brownian motion with p = 0 and q > 0 for the associated local time eL^u_t to be jointly continuous on [0, 1] × R_+^1.

Background

Section 5.1 studies extension of the local-time continuity and exact modulus results to the boundary point 0 in the non-sticky case p = 0 with q > 0. The authors introduce metric criteria involving the kernel Γ and provide several sufficient integrability conditions on the jump measure η—Conditions A, B, and C—that ensure joint continuity on [0, 1] × R_+^1.

The paper does not establish whether these sufficient conditions are sharp, nor does it characterize precisely which Lévy measures η yield joint continuity. The unresolved task is therefore to identify necessary as well as sufficient conditions on η.