# An inverse random source problem for the time fractional diffusion equation driven by a fractional Brownian motion

## TL;DR

This paper investigates the inverse problem of determining statistical properties of a random source in a time fractional diffusion equation driven by fractional Brownian motion, establishing well-posedness and uniqueness results.

## Contribution

It provides the first rigorous analysis of the inverse random source problem for fractional diffusion equations with fractional Brownian motion, including uniqueness and instability characterization.

## Key findings

- The direct problem has a unique mild solution under certain conditions.
- The inverse problem's solution is unique.
- The instability of the inverse problem is characterized.

## Abstract

This paper is concerned with the mathematical analysis of the inverse random source problem for the time fractional diffusion equation, where the source is assumed to be driven by a fractional Brownian motion. Given the random source, the direct problem is to study the stochastic time fractional diffusion equation. The inverse problem is to determine the statistical properties of the source from the expectation and variance of the final time data. For the direct problem, we show that it is well-posed and has a unique mild solution under a certain condition. For the inverse problem, the uniqueness is proved and the instability is characterized. The major ingredients of the analysis are based on the properties of the Mittag--Leffler function and the stochastic integrals associated with the fractional Brownian motion.