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

Author:

Kay Toby1,McKetterick Thomas John1,Giuggioli Luca12ORCID

Affiliation:

1. Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1UB, United Kingdom

2. Bristol Centre for Complexity Sciences, University of Bristol, Bristol, BS8 1UB, United Kingdom

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.

Funder

Engineering and Physical Sciences Research Council

Biotechnology and Biological Sciences Research Council

Publisher

World Scientific Pub Co Pte Ltd

Subject

Condensed Matter Physics,Statistical and Nonlinear Physics

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