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Stochastic persistency of nematic alignment state for the Justh-Krishnaprasad model with additive white noises

Seung-Yeal Ha Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea; Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Republic of Korea Dongnam Ko DeustoTech, University of Deusto, Avenida de las Universidades 24, 48007 Bilbao, Facultad de Ingeniería, Universidad de Deusto, 48007 Bilbao, Basque Country, Spain Woojoo Shim Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea Hui Yu Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, P. R. China

TBD mathscidoc:2205.43002

Mathematical Models and Methods in Applied Sciences, 30, (04), 727-763, 2020.4
We present a stochastic Justh–Krishnaprasad flocking model describing interactions among individuals in a planar domain with their positions and heading angles. The deterministic counterpart of the proposed model describes the formation of nematic alignment in an ensemble of planar particles moving with a unit speed. When the noise is turned off, we show that the nematic alignment state, in which all heading angles are either same or the opposite, is nonlinearly stable using a Lyapunov functional approach. We employed a diameter-like functional via the rearrangement of heading angles in the 2π-interval. In contrast, under the additive noise, a continuous angle configuration will be deviated asymptotically from the nematic state. Nevertheless, in any finite-time interval, we will see that some part of angle configuration will stay close to the nematic state with a positive probability, where we call this phenomenon as stochastic persistency. We provide a quantitative estimate on the probability for stochastic persistency and compare several numerical examples with analytical results.
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@inproceedings{seung-yeal2020stochastic,
  title={Stochastic persistency of nematic alignment state for the Justh-Krishnaprasad model with additive white noises},
  author={Seung-Yeal Ha, Dongnam Ko, Woojoo Shim, and Hui Yu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220520142420999294304},
  booktitle={Mathematical Models and Methods in Applied Sciences},
  volume={30},
  number={04},
  pages={727-763},
  year={2020},
}
Seung-Yeal Ha, Dongnam Ko, Woojoo Shim, and Hui Yu. Stochastic persistency of nematic alignment state for the Justh-Krishnaprasad model with additive white noises. 2020. Vol. 30. In Mathematical Models and Methods in Applied Sciences. pp.727-763. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220520142420999294304.
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