Brownian motion can feel the shape of a drum

TL;DR

This paper demonstrates that Brownian motion, with appropriate drift, can be used to reconstruct any scenery on a torus by extending Fourier-based criteria from discrete to continuous spaces.

Contribution

It generalizes a Fourier coefficient criterion for scenery reconstruction from discrete cycles to continuous spaces like the torus, proving an injectivity property of an infinite Vandermonde matrix.

Findings

Brownian motion can reconstruct any scenery with the right drift.

Fourier coefficient criteria extend from discrete cycles to continuous spaces.

Injectivity of an infinite Vandermonde matrix is established.

Abstract

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.