Elementary Theory of Stochastic Narrow Escape

Abstract

Stochastic narrow escape, as defined in the Preface, consists in the passage of a diffusing particle through a narrow opening in an impermeable wall or the arrival of the trajectory of a diffusion process at a small target. A stochastic narrow escape is a rare event on the time scale of diffusion in the sense that the times between stochastic narrow escapes may be much longer than the times between all other diffusional events and, indeed, they become infinite as the narrow opening or target shrinks to zero. It is therefore practically impossible to explore the high-dimensional parameter space by Brownian-dynamics or other simulations. Very often these rare events are the manifestations of cellular function, such as cross-membrane ionic currents, neuronal signalling, the effective motion of a receptor between obstacles on a membrane, and so on. The time-scale of cellular events is determined by the time-scale of stochastic narrow escapes of molecular diffusion.