The defect technique for partially absorbing and reflecting boundaries: Application to the Ornstein–Uhlenbeck process

Abstract

A particle roaming in a potential in the presence of spatial heterogeneities such as traps or physical barriers is a mathematical representation of a vast number of processes across multiple disciplines, ranging from the dynamics of chemical reactions and the movement of foraging animals to the performance of a financial asset. Among the approaches to tackle such problems, a powerful one is the so-called defect technique. Within a continuous space–time formulation, the technique has been successfully used to analyze diffusion problems with partially absorbing boundaries, but has not been employed to study other types of boundaries. Here we exploit the continuity equation to extend the defect technique to when partially reflecting barriers are present. For each boundary type, we apply the defect formalism to the Ornstein–Uhlenbeck process, recovering known analytical results and presenting new ones.