Calculating first passage densities using path integrals

Many fluctuation-driven systems have important events that occur when a stochastic process encounters a boundary for the first time, e.g. barrier-crossing in chemical kinetics, integrate-and-fire models for neurons, or the triggering of certain options in finance. In the absence of an external potential, the system is governed by the diffusion equation, which can be solved to determine the first passage time densities. When a potential is present, however, it is the considerably more complicated Fokker-Planck-Smoluchowski equation that must be solved. This can only be done numerically except for the simplest systems, and quickly becomes computationally intractable as the number of dimensions increases. In this paper, we use the path integral formulation of stochastic processes to derive an approximate analytical expression for the first passage density for a system diffusing in a potential, valid in the weak noise limit. This rests on a remarkable mapping between the most-probable trajectories in an overdamped stochastic process, and Hamiltonian mechanics in an effective potential. I will demonstrate the approach using simple models in 1-D, and then discuss extensions to higher dimensions.