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

Abstract We study the inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion with Hurst index H ∈ ( 0 , 1 ) {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 sign. Next, combining the properties of Mittag-Leffler function, the complete uniqueness and instability analyses are provided. It is worth mentioning that all the analyses are unified for H ∈ ( 0 , 1 ) {H\in(0,1)} .