  Anomalous subdiffusion characterizes transport in diverse physical systems
and is especially prevalent inside biological cells. In cell biology, the
prevailing model for chemical activation rates has recently changed from the
first passage time (FPT) of a single searcher to the FPT of the fastest
searcher out of many searchers to reach a target, which is called an extreme
statistic or extreme FPT. In this paper, we investigate extreme statistics of
searchers which move by anomalous subdiffusion. We model subdiffusion by a
fractional Fokker-Planck equation involving the Riemann-Liouville fractional
derivative. We prove an explicit and very general formula for every moment of
subdiffusive extreme FPTs and approximate their full probability distribution.
While the mean FPT of a single subdiffusive searcher is infinite, the fastest
subdiffusive searcher out of many subdiffusive searchers typically has a finite
mean FPT. In fact, we prove the counterintuitive result that extreme FPTs of
subdiffusion are faster than extreme FPTs of normal diffusion. Mathematically,
we employ a stochastic representation involving a random time change of a
standard Ito drift-diffusion according to the trajectory of the first crossing
time inverse of a Levy subordinator. A key step in our analysis is generalizing
Varadhan's formula from large deviation theory to the case of subdiffusion,
which yields the short-time distribution of subdiffusion in terms of a certain
geodesic distance.
