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

## TL;DR

This paper establishes sharp convergence rates for a time discretization scheme applied to stochastic time-fractional PDEs with white noise, demonstrating optimal error bounds and validating them through numerical experiments.

## Contribution

The paper provides the first sharp-order error estimates for backward-Euler convolution quadrature applied to stochastic time-fractional PDEs with white noise, including optimal rates in one dimension.

## Key findings

- Error estimate: $O( au^{1-rac{ ext{alpha} imes d}{2}})$ for the discretization
- Optimal convergence rates for stochastic subdiffusion and diffusion-wave problems in 1D
- Numerical examples confirm theoretical error bounds

## Abstract

The stochastic time-fractional equation $\partial_t \psi -\Delta\partial_t^{1-\alpha} \psi = f + \dot W$ with space-time white noise $\dot W$ is discretized in time by a backward-Euler convolution quadrature for which the sharp-order error estimate \[ {\mathbb E}\|\psi(\cdot,t_n)-\psi_n\|_{L^2(\mathcal{O})}^2=O(\tau^{1-\alpha d/2}) \] is established for $\alpha\in(0,2/d)$, where $d$ denotes the spatial dimension, $\psi_n$ the approximate solution at the $n^{\rm th}$ 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.