First-Passage of a Brownian Particle - Escape from a Bounded Potential

Markus Nyberg, Tobias Ambjörnsson, Ludvig Lizana

Published 2015-03-27Version 1

Many different processes depend on the first crossing of a boundary. For example, the time it takes for a protein find its target site on DNA, and the waiting time until a neurone starts firing. Motivated by the lack of tools accessible for computing first passage time densities (FPDTs) for general potentials, we propose a new method based on the Independent Interval Approximation (IIA). We generalise the IIA framework to non-smooth Brownian processes and derive a closed form expression for the FPDT in Laplace space for arbitrary boundary and starting point in one dimension. We apply our results to a Brownian particle in a harmonic potential and we find good agreement with Langevin dynamics simulations for one and two boundaries. We anticipate that our results will have a wide applicability an a number of escape problems.