We study quaternionic stochastic areas processes associated with Brownian motions on the quaternionic rank-one symmetric spaces $\mathbb{H}H^n$ and $\mathbb{H}P^n$. The characteristic functions of fixed-time marginals of these processes are computed and allows for the explicit description of their corresponding large-time limits. We also obtain exact formulas for the semigroup densities of the stochastic area processes using a Doob transform in the former case and the semigroup density of the circular Jacobi process in the latter. For $\mathbb{H}H^n$, the geometry of the quaternionic anti-de Sitter fibration plays a central role , whereas for $\mathbb{H}P^n$, this role is played by the quaternionic Hopf fibration.

中文翻译：

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四元数随机区域

我们在四元数秩一对称空间 $\mathbb{H}H^n$ 和 $\mathbb{H}P^n$ 上研究与布朗运动相关的四元数随机区域过程。计算这些过程的固定时间边际的特征函数，并允许对其相应的大时间限制进行​​明确描述。我们还获得了随机区域过程的半群密度的精确公式，在前一种情况下使用 Doob 变换，在后一种情况下使用圆形雅可比过程的半群密度。对于$\mathbb{H}H^n$，四元反德西特纤维化的几何结构起着核心作用，而对于$\mathbb{H}P^n$，这个作用由四元Hopf 纤维化扮演。