Averaging principle for a slow-fast stochastic nonlinear fractional Schr\"odinger equation

TL;DR

This paper proves an averaging principle for a multiscale stochastic nonlinear fractional Schr"odinger system, showing the slow component converges to an effective equation as scale separation increases.

Contribution

It introduces a novel averaging technique for a complex fractional Schr"odinger system with polynomial nonlinearities and stochastic forcing, extending existing methods to fractional dispersive operators.

Findings

Strong convergence of the slow component to an effective stochastic fractional Schr"odinger equation.

Derivation of the effective drift by averaging over the invariant measure of the fast dynamics.

Handling of limited smoothing and nonlinearities through refined estimates and viscosity approximation.

Abstract

We establish an averaging principle for a structural multiscale stochastic nonlinear fractional Schr\"odinger system on the one-dimensional torus driven by a multiplicative Wiener noise. The slow component is governed by a fractional Schr\"odinger operator with a general polynomial nonlinearity, while the fast component evolves on a shorter time scale and exhibits dissipative diffusion, nonlinear interactions, and stochastic forcing. Under suitable dissipative assumptions, we have shown that, as the scale separation parameter tends to zero, the slow component converges strongly to an effective stochastic fractional Schr\"odinger equation. The effective drift is obtained by averaging the coupling term with respect to the unique invariant measure of the frozen fast dynamics. The proof relies on uniform a priori estimates, ergodicity of the fast equation, H\"older time regularity of the slow component obtained via a vanishing viscosity method, and a Khasminskii-type time discretization argument adapted to fractional dispersive operators. The analysis is technically challenging due to limited smoothing of the fractional Schr\"odinger semigroup and the presence of general polynomial nonlinearities, which are handled through refined estimates and viscosity approximation.