Boundary conditions and trajectories of diffusion processes

Abstract

This article constructs trajectories associated with various boundary conditions for the Smoluchowski equation on an interval. Single-particle diffusion processes are first constructed by taking the diffusion limits of random walks. The diffusion limit gives both boundary conditions which enforce the single-particle constraint and properties of underlying trajectories at those boundaries. Mean-field diffusions are then obtained as limits of sums of single-particle processes. The results help to interpret the application of diffusion models to both ion channels and wider pores that facilitate molecular transport across membranes. Potential applications to Brownian dynamics simulations are discussed.