Renewal theory for Brownian motion across a stochastically gated interface
File(s)
Published version
Author(s)
Bressloff, Paul C
Type
Journal Article
Abstract
Stochastically gated interfaces play an important role in a variety of cellular transport processes,
including diffusion through membrane ion channels and intercellular gap junctions. Most stud ies of stochastically-gated interfaces are based on macroscopic models that track the particle
concentration averaged with respect to different realisations of the gate dynamics. In this paper
we develop a novel probabilistic model of single-particle Brownian motion (BM) through a
stochastically gated interface. We proceed by constructing a renewal equation for one-dimensional
BM with an interface at the origin, which effectively sews together a sequence of BMs on the half line with a totally absorbing boundary at x = 0. Each time the particle is absorbed, the stochastic
process is immediately restarted according to the following rule: if the gate is closed then BM
restarts on the same side of the interface, whereas if the gate is open then BM restarts on either
side of the interface with equal probability. In order to ensure that diffusion restarts in a state that
avoids immediate re-absorption. we assume that whenever the particle reaches the interface it is
instantaneously shifted a distance ϵ from the origin. We explicitly solve the renewal equation for
ϵ > 0 and show how the solution of a corresponding forward Kolmogorov equation is recovered in
the limit ϵ → 0. However, the renewal equation provides a more general mathematical framework
for modelling a stochastically gated interface by explicitly separating the first passage time problem
of detecting the gated interface (absorption) and the subsequent rule for restarting BM. We illus trate this by calculating the non-equilibrium stationary state across an interface in the presence of
stochastic resetting. We conclude by discussing some of the mathematical challenges in extending
the theory to higher-dimensional interfaces
including diffusion through membrane ion channels and intercellular gap junctions. Most stud ies of stochastically-gated interfaces are based on macroscopic models that track the particle
concentration averaged with respect to different realisations of the gate dynamics. In this paper
we develop a novel probabilistic model of single-particle Brownian motion (BM) through a
stochastically gated interface. We proceed by constructing a renewal equation for one-dimensional
BM with an interface at the origin, which effectively sews together a sequence of BMs on the half line with a totally absorbing boundary at x = 0. Each time the particle is absorbed, the stochastic
process is immediately restarted according to the following rule: if the gate is closed then BM
restarts on the same side of the interface, whereas if the gate is open then BM restarts on either
side of the interface with equal probability. In order to ensure that diffusion restarts in a state that
avoids immediate re-absorption. we assume that whenever the particle reaches the interface it is
instantaneously shifted a distance ϵ from the origin. We explicitly solve the renewal equation for
ϵ > 0 and show how the solution of a corresponding forward Kolmogorov equation is recovered in
the limit ϵ → 0. However, the renewal equation provides a more general mathematical framework
for modelling a stochastically gated interface by explicitly separating the first passage time problem
of detecting the gated interface (absorption) and the subsequent rule for restarting BM. We illus trate this by calculating the non-equilibrium stationary state across an interface in the presence of
stochastic resetting. We conclude by discussing some of the mathematical challenges in extending
the theory to higher-dimensional interfaces
Date Issued
2026-03-13
Date Acceptance
2026-02-24
Citation
Journal of Physics A: Mathematical and Theoretical, 2026, 59 (10)
ISSN
1751-8113
Publisher
IOP Publishing
Journal / Book Title
Journal of Physics A: Mathematical and Theoretical
Volume
59
Issue
10
Copyright Statement
© 2026 The Author(s). Published by IOP Publishing Ltd Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
License URL
Publication Status
Published
Article Number
105003
Date Publish Online
2026-03-11