标题： 高维Hegselmann-Krause动力系统的准同步时间的性质 显示英文标题

标题： Properties of Quasi-synchronization Time of High-dimensional Hegselmann-Krause Dynamics

摘要： 一维Hegselmann-Krause（HK）动力学在噪声驱动下的行为已被广泛研究。 先前的研究表明，无论是在有界还是无界的二维空间中，HK动力学都能在有限时间内达到准同步（噪声情况下的同步）。 然而，这一现象在高维空间中是否成立仍不清楚。 本文研究了多维HK模型的准同步随机时间，并揭示了空间的有界性和维度决定了不同的结果。 具体来说，如果空间是有界的，在有限时间内对于所有维度都可以几乎必然达到准同步，而在无界空间中，仅在低维情况下（一维和二维）可以实现准同步。 此外，还证明了不同情况下随机时间的不同可积性。 显示英文摘要

摘要： The behavior of one-dimensional Hegselmann-Krause (HK) dynamics driven by noise has been extensively studied. Previous research has indicated that within no matter the bounded or the unbounded space of one dimension, the HK dynamics attain quasi-synchronization (synchronization in noisy case) in finite time. However, it remains unclear whether this phenomenon holds in high-dimensional space. This paper investigates the random time for quasi-synchronization of multi-dimensional HK model and reveals that the boundedness and dimensions of the space determine different outcomes. To be specific, if the space is bounded, quasi-synchronization can be attained almost surely for all dimensions within a finite time, whereas in unbounded space, quasi-synchronization can only be achieved in low-dimensional cases (one and two). Furthermore, different integrability of the random time of various cases is proved.