Large torus limit of global dynamics of the two-dimensional dispersive Anderson model

TL;DR

This paper investigates the behavior of the two-dimensional dispersive Anderson model on large tori, showing convergence of periodic solutions to full-space solutions using weighted function spaces and analyzing related parabolic models.

Contribution

It introduces a novel approach with weighted function spaces to analyze the large torus limit of the dispersive Anderson model, extending previous well-posedness results.

Findings

Periodic solutions converge to full-space solutions as period increases

Weighted function spaces effectively control noise growth

Analysis includes the parabolic Anderson model as well

Abstract

We continue the study of the two-dimensional dispersive Anderson model (DAM), i.e. the nonlinear Schr\"odinger equation with multiplicative spatial white noise. For this model, global well-posedness on the periodic domain was established by Visciglia and the second author (2023), and global well-posedness on the full space was established by Debussche, Visciglia, and the authors (2024). We show that, under suitable initial conditions and suitable periodization procedure of the noise, the periodic global dynamics of the DAM converges in spaces of local domains to that of the DAM on the full space as the period goes to infinity. In order to control the growth of the noise and obtain a priori bounds for solutions independent of the periodicity, we introduce periodic weights and construct weighted function spaces on periodic domains. In Appendix, we also discuss the same problem for the parabolic Anderson model.