Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise

The stochastic time-fractional equation ${\mathrm{\partial }}_t\psi -\mathrm{\Delta }{\mathrm{\partial }}_t^{1-\alpha }\psi =f+\dot{W}$ with space-time white noise $\dot{W}$ is discretized in time by a backward-Euler convolution quadrature for which the sharp-order error estimate $$\mathbb{E}\mathrm{\parallel }\psi (\cdot ,t_n)-{\psi }_n{\mathrm{\parallel }}_{L^2(\mathcal{O})}^2=O({\tau }^{1-\alpha d\mathrm{/}2})$$ is established for $\alpha \in (0,2\mathrm{/}d)$, where $d$ denotes the spatial dimension, ${\psi }_n$ the approximate solution at the $n^{\mathrm{t}\mathrm{h}}$ time step, and $\mathbb{E}$ the expectation operator. In particular, the result indicates optimal 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.