Renewal theory for Brownian motion across a stochastically gated interface

TL;DR

This paper develops a renewal-theoretic framework for Brownian motion across stochastically gated interfaces, explicitly separating the first-passage event (detection of the gate) from the restart rule. It derives a renewal equation for 1D BM with a gate at the origin, incorporating a small offset ε to prevent immediate reabsorption, and provides closed-form Laplace-space solutions for finite ε that converge to the forward Kolmogorov PDE in the limit ε→0. The fast-switching limit recovers a semipermeable boundary with permeability κ0, and the framework is extended to stochastic resetting, yielding nonequilibrium stationary states. The theory is further generalized to higher-dimensional interfaces, where the renewal equations become surface-integral Fredholm problems on ∂M and connect to Dirichlet-to-Neumann operators, highlighting both analytical structure and numerical challenges for complex geometries.

Abstract

Stochastically gated interfaces play an important role in a variety of cellular diffusion processes. Examples include intracellular transport via stochastically gated ion channels and pores in the plasma membrane of a cell, intercellular transport between cells coupled by stochastically gated gap junctions, and oxygen transport in insect respiration. Most studies 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 use renewal theory to develop a probabilistic model of single-particle Brownian motion (BM) through a stochastically gated interface. We proceed by constructing a renewal equation for 1D 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 $ε\rightarrow 0$. However, the renewal equation provides a more general mathematical framework by explicitly separating the first passage time problem of detecting the gated interface (absorption) and the subsequent rule for restarting BM. We conclude by extending the theory to higher-dimensional interfaces.

Figures (8)

• Figure 1: Brownian particle in ${\mathbb R}$ with a stochastically gated interface at $x=0$. The gate is open (permeable) if $\sigma(t)=1$ and closed (impermeable) if'$\sigma(t)=-1$. Transitions $\sigma(t)\rightarrow -\sigma(t)$ occurs according to a two-state Markov chain with transition rates $\alpha,\beta$.
• Figure 2: Snapping out BM in ${\mathbb R}$ with a semipermeable interface at $x=0$. Each side of the barrier is treated as a partially reflecting boundary. Schematic illustration of a sample trajectory starting at $x_0>0$. Each time the particle is absorbed at $x=0^{\pm}$ an unbiased coin is tossed to determine which side of the barrier BM restarts.
• Figure 3: Brownian particle in ${\mathbb R}$ with a stochastically gated interface at $x=0$. Each side of the barrier is treated as a totally absorbing boundary. Schematic illustration of a sample trajectory starting at $x_0>0$. Each time the particle reaches $x=0$ and the gate is closed, BM restarts on the same side of the barrier and the restart position is taken to be a distance $\epsilon$ from the interface. On the other hand, if the gate is open then BM restarts on either side of the interface with equal probability.
• Figure 4: BM through a stochastically gated interface with stochastic resetting to $\pm \xi$. (The dynamics is extended into two dimensions for ease of visualization.) The particle starts on the right-hand side of the interface and undergoes one reset to $\xi$ before passing through an open gate to the left-hand side. Whilst in this domain the particle reflects off a closed gate and then resets to $-\xi$ etc. Resetting events that cross the membrane are forbidden.
• Figure 5: NESS $\overline{\rho}^{(\epsilon)}(x))$ for 1D BM with a stochastically-gated interface at $x=0$, bulk resetting to $\pm \xi$ at a rate $r$ (see Fig.\ref{['fig4']}), and boundary-induced resetting with a shift of size $\epsilon$. Since $\overline{\rho}^{(\epsilon)}(x)$ is an even function of $x$ we show plots for various values of $\epsilon$ with $x\geq 0$. Other parameters are $D=\xi=1$, $\alpha=\beta=1$ and $r=1$.