Brownian motion can feel the shape of a drum

We study the scenery reconstruction problem on the d-dimensional torus, proving that a criterion on Fourier coefficients obtained by Matzinger and Lember (2006) for discrete cycles applies also in continuous spaces. In particular, with the right drift, Brownian motion can be used to reconstruct any scenery. To this end, we prove an injectivity property of an infinite Vandermonde matrix.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This study focuses on the problem of scenery reconstruction on d-dimensional torus. The researchers proved that the criterion on Fourier coefficients for discrete cycles, discovered by Matzinger and Lember in 2006, also applies in continuous spaces. It is shown that with the right drift, Brownian motion can be used to reconstruct any scenery. The injectivity property of an infinite Vandermonde matrix is also proven.