Solutions of first-passage time problems: A biscaling approach

We study the first-passage time (FPT) problem for widespread recurrent processes in confined though large systems and present a comprehensive framework for characterizing the FPT distribution over many timescales. We find that the FPT statistics can be described by two scaling functions: one corresponds to the solution for an infinite system, and the other describes a scaling that depends on system size. We find a universal scaling relationship for the FPT moments 〈t^{q}〉 with respect to the domain size and the source-target distance. This scaling exhibits a transition at q_{c}=θ, where θ is the persistence exponent. For low-order moments with q<q_{c}, convergence occurs towards the moments of an infinite system. In contrast, the high-order moments, q>q_{c}, can be derived from an infinite density function. The presented uniform approximation, connecting the two scaling functions, provides a description of the first-passage time statistics across all timescales. We extend the results to include diffusion in a confining potential in the high-temperature limit, where the potential strength takes the place of the system's size as the relevant scale. This study has been applied to various mediums, including a particle in a box, two-dimensional wedge, fractal geometries, non-Markovian processes, and the nonequilibrium process of resetting.