Convergence of a $\theta$-scheme to solve the stochastic nonlinear Schr\"odinger equation with Stratonovich noise

TL;DR

This paper introduces a $ heta$-scheme for discretizing the stochastic nonlinear Schrödinger equation with Stratonovich noise, establishing convergence and optimal order accuracy in the strong local sense.

Contribution

The paper develops a novel $ heta$-scheme for the stochastic Schrödinger equation and proves its convergence with an optimal order, including uniform bounds for the Hamiltonian.

Findings

Uniform bounds for the discrete Hamiltonian are established.

The scheme achieves an optimal convergence order of 1 in the strong local sense.

Convergence in probability towards a mild solution is verified.

Abstract

We propose a $\theta$-scheme to discretize the $d$-dimensional stochastic cubic Schr\"odinger equation in Stratono\-vich sense. A uniform bound for the Hamiltonian of the discrete problem is obtained, which is a crucial property to verify the convergence in probability towards a mild solution. Furthermore, based on the uniform bounds of iterates in ${\mathbb H}^2(\mathcal{O})$ for $\mathcal{O}\subset\mathbb{R}^{1}$, the optimal convergence order 1 in strong local sense is obtained.