Strong Convergence Rate of Finite Difference Approximations for Stochastic Cubic Schr\"odinger Equations

TL;DR

This paper establishes a strong convergence rate for finite difference methods applied to stochastic cubic Schrödinger equations, utilizing exponential integrability and Lyapunov functionals to ensure solution boundedness and stability.

Contribution

It introduces a novel approach combining Lyapunov functionals and exponential integrability to analyze convergence of finite difference schemes for stochastic Schrödinger equations.

Findings

Proves strong convergence rate of finite difference approximations.

Establishes exponential integrability of solutions.

Demonstrates continuous dependence on initial data.

Abstract

In this paper, we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic Schr\"odinger equations driven by a multiplicative $Q$-Wiener process. Beyond the uniform boundedness of moments for high order derivatives of the exact solution, the key requirement of our approach is the exponential integrability of both the exact and numerical solutions. By constructing and analyzing a Lyapunov functional and its discrete correspondence, we derive the uniform boundedness of moments for high order derivatives of the exact solution and the first order derivative of the numerical solution, which immediately yields the well-posedness of both the continuous and discrete problems. The latter exponential integrability is obtained through a variant of a criterion given by [Cox, Hutzenthaler and Jentzen, arXiv:1309.5595]. As a by-product of this exponential integrability, we prove that the exact and numerical solutions depend continuously on the initial data and obtain a large deviation-type result on the dependence of the noise with first order strong convergence rate.