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

Extreme statistics of anomalous subdiffusion following a fractional Fokker-Planck equation: Subdiffusion is faster than normal diffusion