Analytic theory of extrema. I. Asymptotic theory for Fokker–Planck processes

We develop a simple method that yields analytic results for the mean first passage time and mean extreme value of Fokker–Planck processes in the asymptotic regime. The results are very sensitive to the detailed structure of the drift and diffusion terms in the Fokker–Planck equation. We apply our method to a variety of systems with additive fluctuations such as a Brownian particle. We also analyze processes with multiplicative fluctuations and obtain explicit results for the extreme value statistics of the energy envelope of a lightly damped harmonic oscillator and of the Verhulst and Gompertz population models.