Spectral-capacitary bounds for homological recovery from reflected Brownian trajectories

Abstract

Can the topology of an unknown reflecting domain be recovered from one reflected Brownian trajectory, and how long must it be observed? We formulate this as a minimax homological inference problem based on the reflected Wiener sausage. The optimal observation time is, up to constants, the sum of three terms: a spectral-access or burn-in time set by the Neumann spectral gap; an inverse-capacity detection time for small topological features, showing that Brownian capacity rather than volume controls discovery; and a logarithmic factor in the number of features to be resolved. Matching minimax lower bounds prove that all three contributions are necessary. The proof combines a finite-target spectral hitting estimate, expressed through killed eigenvalues, with a uniform small-hole eigenvalue theorem comparing the principal killed eigenvalue of an epsilon-scale feature with its Brownian capacity, uniformly over interior and boundary targets. The resulting framework distinguishes full reconstruction, obtained from an intrinsic epsilon-net and yielding two-sided homology isomorphisms, from faster feature detection by witness regions, which gives the corresponding surjective homological recovery.