Averaging principle for a slow-fast stochastic nonlinear fractional Schrödinger equation

Abstract

We establish an averaging principle for a structural multiscale stochastic nonlinear fractional Schrödinger system on the one-dimensional torus driven by a multiplicative Wiener noise. The slow component is governed by a fractional Schrödinger 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 Schrödinger 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, Hölder 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 Schrödinger semigroup and the presence of general polynomial nonlinearities, which are handled through refined estimates and viscosity approximation.