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PreprintCC BY 4.0arXiv

Stochastic Flows with Strong Shear - Part I: Strong Completeness and Set Attractors

  • Dennis Chemnitz
  • Maximilian Engel
  • Michael Scheutzow
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Abstract

We study strong completeness and the existence of a set attractor for the stochastic flows induced by a class of two-dimensional stochastic differential equations resembling a planar Ornstein-Uhlenbeck process with an additional radius-dependent rotational drift term. Our main results give both sufficient and necessary conditions for the stochastic flows to be strongly complete and for the existence of set attractors. In particular, we demonstrate that if the derivative of the angular velocity ρ(r) with respect to the radius r satisfies |ρ'(r)| ≥ K2 r3, for large r and some universal constant K2, the diameter of a compact set can grow exponentially fast with positive probability. Furthermore, we show that for |ρ'(r)|≥ r3+ε, ε>0, with probability one, exceptional initial conditions diverge to infinity in finite time, ruling out the existence of a global stochastic flow.

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Cite

BibTeXRIS
BibTeX
@misc{chemnitz2026stochastic,
  title = {{Stochastic Flows with Strong Shear - Part I: Strong Completeness and Set Attractors}},
  author = {Dennis Chemnitz and Maximilian Engel and Michael Scheutzow},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06321},
  eprint = {2610.06321},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06321},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Stochastic Flows with Strong Shear - Part I: Strong Completeness and Set Attractors
AU  - Dennis Chemnitz
AU  - Maximilian Engel
AU  - Michael Scheutzow
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06321
UR  - https://arxiv.org/abs/2610.06321
AB  - We study strong completeness and the existence of a set attractor for the stochastic flows induced by a class of two-dimensional stochastic differential equations resembling a planar Ornstein-Uhlenbeck process with an additional radius-dependent rotational drift term. Our main results give both sufficient and necessary conditions for the stochastic flows to be strongly complete and for the existence of set attractors. In particular, we demonstrate that if the derivative of the angular velocity ρ(r) with respect to the radius r satisfies |ρ'(r)| ≥ K_2 r^3, for large r and some universal constant K_2, the diameter of a compact set can grow exponentially fast with positive probability. Furthermore, we show that for |ρ'(r)|≥ r^3+ε, ε>0, with probability one, exceptional initial conditions diverge to infinity in finite time, ruling out the existence of a global stochastic flow.
ER  -