  Recent advancements in understanding the impulse response of the first
arrival position (FAP) channel in molecular communication (MC) have illuminated
its Shannon capacity. While Lee et al. shed light on FAP channel capacities
with vertical drifts, the zero-drift scenario remains a conundrum, primarily
due to the challenges associated with the heavy-tailed Cauchy distributions
whose first and second moments do not exist, rendering traditional mutual
information constraints ineffective. This paper unveils a novel
characterization of the zero drift FAP channel capacity for both 2D and 3D.
Interestingly, our results reveal a 3D FAP channel capacity that is double its
2D counterpart, hinting at a capacity increase with spatial dimension growth.
Furthermore, our approach, which incorporates a modified logarithmic constraint
and an output signal constraint, offers a simplified and more intuitive formula
(similar to the well-known Gaussian case) for estimating FAP channel capacity.
