An inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion

TL;DR

This paper investigates the inverse problem of identifying sources in a time-space fractional diffusion equation driven by fractional Brownian motion, providing regularity, uniqueness, and instability results applicable for all Hurst indices.

Contribution

It introduces a novel estimate and a unified analysis framework for the inverse source problem in fractional diffusion equations driven by fractional Brownian motion.

Findings

Reconstruction scheme for source terms $f$ and $g$ up to sign

Complete uniqueness and instability analyses

Unified approach valid for all Hurst indices $H\

Abstract

We study the inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion with Hurst index $H\in(0,1)$. With the aid of a novel estimate, by using the operator approach we propose regularity analyses for the direct problem. Then we provide a reconstruction scheme for the source terms $f$ and $g$ up to the sign. Next, combining the properties of Mittag-Leffler function, the complete uniqueness and instability analyses are provided. It's worth mentioning that all the analyses are unified for $H\in(0,1)$.