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 [Formula: see text]-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.

中文翻译：

---

具有加性白噪声的 Justh-Krishnaprasad 模型的向列对齐状态的随机持久性

我们提出了一个随机 Justh-Krishnaprasad 植绒模型，该模型描述了平面域中个体之间的相互作用及其位置和航向角。所提出模型的确定性对应物描述了在以单位速度移动的平面粒子集合中向列排列的形成。当噪声关闭时，我们使用 Lyapunov 函数方法证明了所有航向角相同或相反的向列对齐状态是非线性稳定的。我们通过在[公式：见文本]-区间中重新排列航向角来使用类直径泛函。相反，在加性噪声下，连续角度配置将逐渐偏离向列状态。然而，在任何有限时间间隔内，我们将看到角度配置的某些部分将以正概率保持接近向列状态，我们将这种现象称为随机持久性。我们提供了对随机持续性概率的定量估计，并将几个数值示例与分析结果进行了比较。