First-passage times to a fractal boundary: Local persistence exponent and its log-periodic oscillations

We investigate the statistics of the first-passage time (FPT) to a fractal self-similar boundary of the Koch snowflake. When the starting position is fixed near the absorbing boundary, the FPT distribution exhibits an apparent power-law decay over a broad range of timescales, culminated by an exponential cutoff. By extensive Monte Carlo simulations, we compute the local persistence exponent of the survival probability and reveal its log-periodic oscillations in time due to self-similarity of the boundary. The effect of the starting point on this behavior is analyzed in depth. Theoretical bounds on the survival probability are derived from the analysis of diffusion in a circular sector. Physical rationales for the refined structure of the survival probability are presented.