First-return time: how long until a random walker revisits its starting position. For ordinary random walks, the answer depends on both the waiting-time distribution (how long between steps) and the jump-size distribution (how far each step goes). The two distributions jointly determine the return statistics.
In fractional kinetics — continuous-time random walks with Mittag-Leffler waiting times — this coupling breaks. The first-return time density depends exclusively on the waiting-time distribution and is completely independent of the jump-size distribution. Whether the jumps are Gaussian, heavy-tailed, finite-variance, or infinite-variance, the return statistics are identical. Only the temporal structure matters; the spatial structure is invisible.
The result holds for both Markovian and non-Markovian settings, with exact expressions provided for each. The ordering matters — jump-first versus wait-first — producing different densities, but in neither case does the jump distribution appear.
The mechanism: fractional diffusion creates a temporal bottleneck so severe that spatial details are irrelevant. The waiting-time distribution controls the long pauses between steps, and these pauses dominate the return time so completely that the geometry of the walk — how far each step reaches, how the walker explores space — contributes nothing to the return statistics. The walker returns when the clock allows, not when the geometry permits. Time, not space, is the binding constraint.