Polynomial mixing for the weakly damped stochastic nonlinear Schrödinger equation on the whole space

@misc{nersesyan2026polynomial,
  title = {{Polynomial mixing for the weakly damped stochastic nonlinear Schrödinger equation on the whole space}},
  author = {Vahagn Nersesyan and Tianyi Pan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08935},
  eprint = {2609.08935},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08935},
}

TY  - UNPB
TI  - Polynomial mixing for the weakly damped stochastic nonlinear Schrödinger equation on the whole space
AU  - Vahagn Nersesyan
AU  - Tianyi Pan
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08935
UR  - https://arxiv.org/abs/2609.08935
AB  - We consider the weakly damped stochastic nonlinear Schrödinger (NLS) equation on the real line, driven by a noise that is white in time and smooth in space. Assuming that the noise is sufficiently non-degenerate, we prove that the equation has a unique stationary measure in the class of probability measures concentrated on H^2, and establish polynomial mixing in the dual-Lipschitz metric over H^1. We do not impose any restriction on the size of the damping. The proof is based on a coupling argument, whose key ingredient is a Foiaş-Prodi-type estimate in the H^1-norm, derived by means of a Lyapunov functional adapted to the linearized NLS dynamics. To compensate for the loss of compactness, we combine this estimate with a truncated Poincaré inequality and a space-time weight function quantifying the spatial decay of solutions.
ER  -