A unified convergence analysis for the fractional diffusion equation driven by fractional Gaussion noise with Hurst index $Hin(0,1)$

Here, we provide a unified framework for numerical analysis of stochastic nonlinear fractional diffusion equation driven by fractional Gaussian noise with Hurst index $Hin(0,1)$. A novel estimate of the second moment of the stochastic integral with respect to fractional Brownian motion is constructed, which greatly contributes to the regularity analyses of the solution in time and space for $Hin(0,1)$. Then we use spectral Galerkin method and backward Euler convolution quadrature to discretize the fractional Laplacian and Riemann-Liouville fractional derivative, respectively. The sharp error estimates of the built numerical scheme are also obtained. Finally, the extensive numerical experiments verify the theoretical results.