# The Calder\'on problem for a space-time fractional parabolic equation

## TL;DR

This paper investigates an inverse problem for a space-time fractional parabolic operator, establishing unique determination of an unknown potential from exterior measurements using advanced mathematical techniques.

## Contribution

It introduces a novel approach to uniquely identify the potential in a fractional parabolic equation using Runge approximation and Carleman estimates, extending inverse problem theory.

## Key findings

- Unique determination of potential Q from exterior data
- Development of Carleman estimates for fractional parabolic operators
- Constructive single measurement results for potential recovery

## Abstract

In this article we study an inverse problem for the space-time fractional parabolic operator $(\partial_t-\Delta)^s+Q$ with $0<s<1$ in any space dimension. We uniquely determine the unknown bounded potential $Q$ from infinitely many exterior Dirichlet-to-Neumann type measurements. This relies on Runge approximation and the dual global weak unique continuation properties of the equation under consideration. In discussing weak unique continuation of our operator, a main feature of our argument relies on a Carleman estimate for the associated fractional parabolic Caffarelli-Silvestre extension. Furthermore, we also discuss constructive single measurement results based on the approximation and unique continuation properties of the equation.