An encounter-based approach for restricted diffusion with a gradient drift

TL;DR

This paper introduces an encounter-based spectral method to analyze restricted diffusion with gradient drift, providing insights into how external forces influence boundary interactions and reaction kinetics.

Contribution

It extends the Dirichlet-to-Neumann operator framework to include gradient drift, enabling spectral analysis of diffusion with boundary reactions under external forces.

Findings

Spectral decomposition of the full propagator for diffusion with drift.

Impact of constant drift on boundary local time statistics.

Method applicable to various diffusion-influenced reaction scenarios.

Abstract

We develop an encounter-based approach for describing restricted diffusion with a gradient drift towards a partially reactive boundary. For this purpose, we introduce an extension of the Dirichlet-to-Neumann operator and use its eigenbasis to derive a spectral decomposition for the full propagator, i.e., the joint probability density function for the particle position and its boundary local time. This is the central quantity that determines various characteristics of diffusion-influenced reactions such as conventional propagators, survival probability, first-passage time distribution, boundary local time distribution, and reaction rate. As an illustration, we investigate the impact of a constant drift onto the boundary local time for restricted diffusion on an interval. More generally, this approach accesses how external forces may influence the statistics of encounters of a diffusing particle with the reactive boundary.