Recovering semipermeable barriers from reflected Brownian motion

TL;DR

This paper investigates how to recover the shape and location of semipermeable barriers in a planar domain by analyzing reflected Brownian motion data, revealing different regimes and rates of recovery depending on observation and sampling parameters.

Contribution

It introduces a comprehensive analysis of recovery regimes for semipermeable barriers, including explicit algorithms and regime-dependent convergence rates based on observation time and sampling frequency.

Findings

Recovery rate is polynomial for fixed-frequency data.

Recovery rate is exponential in high-frequency regime.

Barrier curvature influences convergence in fixed-frequency case.

Abstract

We study the recovery of one-dimensional semipermeable barriers for a stochastic process in a planar domain. The considered process acts like Brownian motion when away from the barriers and is reflected upon contact until a sufficient but random amount of interaction has occurred, determined by the permeability, after which it passes through. Given a sequence of samples, we wonder when one can determine the location and shape of the barriers. This paper identifies several different recovery regimes, determined by the available observation period and the time between samples, with qualitatively different behavior. The observation period $T$ dictates if the full barriers or only certain pieces can be recovered, and the sampling rate significantly influences the convergence rate as $T\to \infty$. This rate turns out polynomial for fixed-frequency data, but exponentially fast in a high-frequency regime. Further, the environment's impact on the difficulty of the problem is quantified using interpretable parameters in the recovery guarantees, and is found to also be regime-dependent. For instance, the curvature of the barriers affects the convergence rate for fixed-frequency data, but becomes irrelevant when $T\to \infty$ with high-frequency data. The results are accompanied by explicit algorithms, and we conclude by illustrating the application to real-life data.