Extreme values of infinite-measure processes.

Abstract

We study the statistics of the maximum and minimum of a set of N random variables whose dynamical and statistical properties fall within the scope of infinite ergodic theory. These nonstationary yet recurrent systems are described, in the long-time limit, by a non-normalizable infinite invariant density. Extreme events in such systems emerge in a joint limit where the observation time t is long and the number of variables N is large. We show that the resulting extreme-value statistics is controlled by the return exponent α and the infinite invariant measure, and therefore departs from the classical Fréchet, Gumbel, and Weibull universality classes. We illustrate the theory for weakly chaotic intermittent maps, overdamped diffusion in an asymptotically flat potential, and a stochastic model of subrecoil laser cooling, and show how measurements of extremes can be used to infer the infinite-density structure.