Convergence of an operator splitting scheme for abstract stochastic evolution equations
Joshua L Padgett, Qin Sheng
TL;DR
This work analyzes the convergence of a Lie-Trotter operator splitting for stochastic semilinear evolution equations in a Hilbert space, including both the continuous and spatially discretized problems. By establishing stability of the splitting map and performing a stochastic Taylor-based consistency analysis under precise noise-regularity assumptions, the authors show that the strong convergence rate is optimal at $\frac{1}{2}$, governed by the regularity parameter $\beta$ of the noise. The key contribution is a rigorous mean-square convergence bound $\mathbb{E}\| (S^n-T^n)(u_0)\|^2 \le C h^{\beta}$, implying a global rate of $\beta/2$ and clarifying how noise type (e.g., trace-class vs white noise) affects the achievable accuracy. The results provide guidance for implementing operator splitting schemes in stochastic PDE contexts and extend to spatial discretizations under the same framework.
Abstract
In this paper we study the convergence of a Lie-Trotter operator splitting for stochastic semi-linear evolution equations in a Hilbert space. The abstract Hilbert space setting allows for the consideration of convergence of the approximation for both the original and spatially discretized problems. It is known that the strong convergence of this scheme is classically of half-order, at best. We demonstrate that this is in fact the optimal order of convergence in the proposed setting, with the actual order being dependent upon the regularity of noise collected from applications.