Global well-posedness and scattering for energy-critical Schrodinger equations with conservative Marcus noise
Abstract
This work establishes global well-posedness for the three-dimensional defocusing energy-critical nonlinear Schrödinger equation with conservative linear Marcus noise. For arbitrary deterministic energy-space data, we construct a probabilistically strong solution, unique in the critical spacetime class, and prove convergence of finite-activity approximations. The spatial profiles need only be bounded and Lipschitz, while the centered pure-jump Lévy noise is assumed to have finite second moment. If the temporal noise coefficient is square integrable on the half-line, the solution scatters almost surely, with uniform mean-square tail convergence and no smallness restriction. The solution conserves mass and satisfies an exact energy balance. The proof combines deterministic energy-critical stability with Poisson Strichartz estimates; convergence of the interaction-representation martingale gives the critical spacetime control needed for continuation and scattering.
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BibTeX
@misc{liu2026global,
title = {{Global well-posedness and scattering for energy-critical Schrodinger equations with conservative Marcus noise}},
author = {Bin Liu and Fan Xu and Bo Yang and Lei Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37534},
eprint = {2609.37534},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37534},
}
RIS
TY - UNPB
TI - Global well-posedness and scattering for energy-critical Schrodinger equations with conservative Marcus noise
AU - Bin Liu
AU - Fan Xu
AU - Bo Yang
AU - Lei Zhang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37534
UR - https://arxiv.org/abs/2609.37534
AB - This work establishes global well-posedness for the three-dimensional defocusing energy-critical nonlinear Schrödinger equation with conservative linear Marcus noise. For arbitrary deterministic energy-space data, we construct a probabilistically strong solution, unique in the critical spacetime class, and prove convergence of finite-activity approximations. The spatial profiles need only be bounded and Lipschitz, while the centered pure-jump Lévy noise is assumed to have finite second moment. If the temporal noise coefficient is square integrable on the half-line, the solution scatters almost surely, with uniform mean-square tail convergence and no smallness restriction. The solution conserves mass and satisfies an exact energy balance. The proof combines deterministic energy-critical stability with Poisson Strichartz estimates; convergence of the interaction-representation martingale gives the critical spacetime control needed for continuation and scattering.
ER -