The narrow escape problem - A short review of recent results

The narrow escape problem in diffusion theory, which goes back to Lord Rayleigh, is to calculate the mean first passage time, also called the narrow escape time (NET), of a Brownian particle to a small absorbing window on the otherwise reflecting boundary of a bounded domain. The renewed interest in the NET problem is due to its relevance in molecular biology and biophysics. The small window often represents a small target on a cellular membrane, such as a protein channel, which is a target for ions, a receptor for neurotransmitter molecules in a neuronal synapse, a narrow neck in the neuronal spine, which is a target for calcium ions, and so on. The leading order singularity of the Neumann function for a regular domain strongly depends on the geometric properties of the boundary. It can give a smaller contribution than the regular part to the absorption flux through the small window when it is located near a boundary cusp. We review here recent results on the dependence of the absorption flux on the geometric properties of the domain and thus reveal geometrical features that can modulate the flux. This indicates a possible way to code information physiologically.