On the splitting method for some complex-valued quasilinear equations

TL;DR

This paper analyzes the convergence speed of the splitting method applied to complex-valued stochastic evolution equations, including parabolic quasi-linear and Schr"odinger equations, in Sobolev spaces.

Contribution

It extends the splitting method analysis to complex-valued equations, providing convergence rates in Sobolev spaces for a broad class of stochastic evolution equations.

Findings

Convergence rates established for complex-valued stochastic equations

Applicable to both parabolic quasi-linear and Schr"odinger equations

Provides theoretical foundation for numerical approximation methods

Abstract

Using the approach of the splitting method developed by I. Gy\"ongy and N. Krylov for parabolic quasi linear equations, we study the speed of convergence for general complex-valued stochastic evolution equations. The approximation is given in general Sobolev spaces and the model considered here contains both the parabolic quasi-linear equations under some (non strict) stochastic parabolicity condition as well as linear Schr\"odinger equations