An inverse random source problem of sub-diffusion equations with fractional Brownian motion on an unbounded domain

This paper concerns both the direct and the inverse source problems for the stochastic time fractional unbounded diffusion equation driven by a fractional Brownian motion. Regarding the direct problem, the well-posedness is established and the regularity of the solution is characterized. For the inverse problem, based on the variance of the temporal data on one fixed spatial point, the uniqueness and instability are investigated on the determination of the Fourier modes of the diffusion coefficient in the random source. To reconstruct the time-dependent source function from its phaseless Fourier modes, the PhaseLift algorithm is employed to tackle the phase retrieval problem. The effectiveness of the proposed method is demonstrated through numerical examples.