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

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