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

Abstract

Here, we provide a unified framework for numerical analysis of stochastic nonlinear fractional diffusion equation driven by fractional Gaussian noise with Hurst index H ∈ (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 H ∈ (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.

Numerical Approximation for Stochastic Nonlinear Fractional Diffusion Equation Driven by Rough Noise

“…But for the case H ∈ (0, 1 2 ) (called as "rough noise"), the existing discussions seem to be few. In [10], the authors propose numerical analyses about the second-order stochastic differential equation driven by spatial fractional Gaussian noise with H ∈ (0, 1 2 ); the reference [23] uses the equivalence of different fractional Sobolev spaces and the assumption τ < τ * (τ * depends on the spatial discretization) to provide a unified strong convergence analysis for fractional stochastic partial differential equation driven by fractional cylinder noise with H ∈ (0, 1); in [9], the authors propose the regularity estimates and the corresponding numerical analyses about the stochastic evolution equation driven by fractional Brownian sheet with H 1 ∈ (0, 1 2 ) and H 2 = 1 2 . In this paper, we focus on the fractional diffusion equation driven by fractional Brownian sheet with Hurst parameters H 1 , H 2 ∈ (0, 1 2 ] numerically, which are both rough in the temporal and spatial directions.…”