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PreprintCC BY 4.0arXiv

Unique ergodicity of projective cocycles over the 2D stochastic Navier-Stokes equations

  • Sam Punshon-Smith
  • Tommaso Rosati
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Abstract

We consider the linear cocycles generated by the linearized vorticity equation and by passive scalar advection diffusion, both driven by the two dimensional stochastic Navier-Stokes flow on the torus with non degenerate additive forcing, and we prove uniqueness of the stationary measures for the projective process associated to such linear dynamics. The proof relies on a localized asymptotic strong coupling construction, and on the non-degeneracy of the Malliavin matrix of the projective process. Establishing this non-degeneracy for passive scalar advection poses additional challenges, and requires a proof (based on Cameron-Martin analyticity arguments) that generically the passive scalar is nowhere one-dimensional.

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Cite

BibTeXRIS
BibTeX
@misc{punshonsmith2026unique,
  title = {{Unique ergodicity of projective cocycles over the 2D stochastic Navier-Stokes equations}},
  author = {Sam Punshon-Smith and Tommaso Rosati},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01697},
  eprint = {2609.01697},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01697},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Unique ergodicity of projective cocycles over the 2D stochastic Navier-Stokes equations
AU  - Sam Punshon-Smith
AU  - Tommaso Rosati
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01697
UR  - https://arxiv.org/abs/2609.01697
AB  - We consider the linear cocycles generated by the linearized vorticity equation and by passive scalar advection diffusion, both driven by the two dimensional stochastic Navier-Stokes flow on the torus with non degenerate additive forcing, and we prove uniqueness of the stationary measures for the projective process associated to such linear dynamics. The proof relies on a localized asymptotic strong coupling construction, and on the non-degeneracy of the Malliavin matrix of the projective process. Establishing this non-degeneracy for passive scalar advection poses additional challenges, and requires a proof (based on Cameron-Martin analyticity arguments) that generically the passive scalar is nowhere one-dimensional.
ER  -