Exact uniform modulus and law of the iterated logarithm at the boundary

Determine an exact uniform modulus of continuity on [0, 1] × R_+^1 and an exact law of the iterated logarithm at 0 for the suitably normalized local time bL^u_t associated with the p = 0, q = 0 Feller Brownian motion in Example 1, where dη(z) = z^{-(1+β)} dz with 0 < β < 1.

Background

Section 5.2 treats the case p = 0 and q = 0, in which the usual normalization of the local time at 0 fails because the potential density u_1(y, 0) vanishes. The authors define a different normalization bL^u_t for u ∈ [0, 1] and show joint continuity for the stable-subordinator example dη(z) = z^{-(1+β)} dz, 0 < β < 1.

For this example, the paper derives upper Hölder-type and modulus bounds, but does not identify the sharp uniform modulus of continuity or the precise law of the iterated logarithm at the boundary point 0. These remain unresolved even after joint continuity has been established.