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.
From: Tristan Guillaume [view email] [via CCSD proxy]
[v1]
Mon, 22 Jun 2026 08:37:15 UTC (1,396 KB)
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