Inverse problems for a fractional conductivity equation

TL;DR

This paper establishes unique determination of an unknown fractional conductivity within a domain using exterior measurements, extending previous results to both single and multiple measurement scenarios, and relates the problem to fractional Schrödinger equations and random walks.

Contribution

It introduces a novel approach to inverse problems for fractional conductivity equations, including reduction to fractional Schrödinger equations and applications to random walk models.

Findings

Global uniqueness for fractional conductivity inverse problems

Extension to single measurement cases

Connection to fractional Schrödinger equations and random walks

Abstract

This paper shows global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, possibly disjoint subsets of the exterior. Both the cases of infinitely many measurements and a single measurement are addressed. The results are based on a reduction from the fractional conductivity equation to the fractional Schr\"odinger equation, and as such represent extensions of previous works. Moreover, a simple application is shown in which the fractional conductivity equation is put into relation with a long jump random walk with weights.