Sharp Convergence Rates of Time Discretization for Stochastic Time-fractional PDEs Subject to Additive Space-time White Noise

The stochastic time-fractional equation ∂ _t Ψ - Δ∂ _t ^1-α Ψ = f + W˙ with space-time white noise W˙ is discretized in time by a backward-Euler convolution quadrature for which the sharp-order error estimate (E||Ψ(·, t _n ) - Ψ _n || _{L ² (O)} ² )) ^1/2 = O(τ ^{1/2 - αd/4} ) is established for α ∈ (0, 2/d), where d denotes the spatial dimension, Ψ _n the approximate solution at the nth time step, and E the expectation operator. In particular, the result indicates sharp convergence rates of numerical solutions for both stochastic subdiffusion and diffusion-wave problems in one spatial dimension. Numerical examples are presented to illustrate the theoretical analysis.