  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.

Analysis of PDEs