We present schemes for simulating Brownian bridges on complete and connected Lie groups and homogeneous spaces. We use this to construct an estimation scheme for recovering an unknown left- or right-invariant Riemannian metric on the Lie group from samples. We subsequently show how pushing forward the distributions generated by Brownian motions on the group results in distributions on homogeneous spaces that exhibit non-trivial covariance structure. The pushforward measure gives rise to new parametric families of distributions on commonly occurring spaces such as spheres and symmetric positive tensors. We extend the estimation scheme to fit these distributions to homogeneous space-valued data. We demonstrate both the simulation schemes and estimation procedures on Lie groups and homogenous spaces, including $\SPD(3) = \GL_+(3)/\SO(3)$ and $\mathbb S^2 = \SO(3)/\SO(2)$.

翻译：我们提出在完整和相连的Lie群和同质空间模拟Brownian桥的模拟计划,我们利用这个计划为从样本中恢复Lile群的未知左位或右位异差Riemannian测量仪建立一个估算计划,随后我们展示了推展Brownian运动在小组上产生的分布是如何在显示非三角共变结构的同质空间上进行分布的。推向措施导致在诸如球体和正对称正数振幅等常见空间上产生新的分布的参数组别。我们扩展了估算计划,以适应这些分布,以适应同质空间价值数据。我们展示了关于Lie组和同质空间的模拟计划和估算程序,包括$\SPD(3) =\GL ⁇ (3)/\SO(3)和$\mathbbs S%2 =\SO(3)/\SO(2)$。