Direct and inverse source problems for a stochastic Riemann-Liouville time-fractional equation with a singular spatial potential

Direct and inverse source problems are studied for a stochastic Riemann--Liouville time-fractional equation with a singular spatial potential. The spatial operator contains a negative potential of the form \(-μ |x|^β\), where \(β∈[-2,+∞)\), and is defined rigorously through a coercive symmetric bilinear form. The spectral structure of the associated elliptic operator yields a modal representation of the projected evolution equations. The Riemann--Liouville initial conditions generate explicit leading singularities in the homogeneous solution; after separating these singular terms, continuity and energy estimates are obtained for the corresponding remainder. For the zero-initial nonhomogeneous stochastic problem, mean-square regularity estimates follow from the spectral representation, Mittag--Leffler kernel bounds and Itô's isometry. The inverse analysis uses terminal statistical observations, following the established spectral moment approach. The terminal mean identifies the deterministic source on its visible spectral modes, whereas the terminal modal variances determine the squared modal intensities of the stochastic source when the temporal stochastic profile is nontrivial. The results characterize identifiable deterministic modes, recovery of stochastic squared modal intensities, and the stability and regularized reconstruction of these quantities. Spectral Tikhonov filtering is applied to the deterministic source, and nonnegative constrained Tikhonov filtering is applied to the stochastic modal intensity. Three-dimensional numerical experiments are presented to illustrate the instability of direct inversion and the effectiveness of the proposed regularization methods.