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.

翻译：本文研究平面区域内随机过程的一维半透性障碍恢复问题。所考虑过程在远离障碍时表现为布朗运动，接触障碍时发生反射，直至达到由渗透性决定的随机交互量后穿透障碍。给定观测样本序列，我们探究在何种条件下能够确定障碍的位置与形状。本文识别了由观测时长与采样间隔决定的多种不同恢复机制，这些机制表现出本质不同的行为特征。观测时长$T$决定了能够恢复完整障碍还是仅能恢复特定片段，而采样频率显著影响$T\to \infty$时的收敛速率：固定频率数据的收敛速度为多项式阶，而高频数据可实现指数级收敛。此外，研究通过可解释参数量化了环境因素对问题难度的影响，发现这种影响同样具有机制依赖性。例如障碍曲率会影响固定频率数据的收敛速率，但在高频数据$T\to \infty$时变得无关紧要。研究结果均配有显式算法说明，最后通过实际数据案例展示了方法的应用。