发表机构
University of Warwick; Florida State University（华威大学; 佛罗里达州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了带诱导度量的地标形状空间在一般度量（含退化度量）下是随机完备的，即黎曼布朗运动对所有时间存在，从而排除了初始不同地标碰撞的可能性，为形状随机建模提供了理论基础。
AI 中文摘要
地标形状空间源于$n$个不同地标的构型，其中刚性运动被去除且尺度被归一化。当配备从微分同胚群上的右不变度量继承而来的足够正则的度量时，形状空间从微分同胚的测地线继承测地完备性。我们现在回答相应的随机问题：地标形状空间上的黎曼布朗运动是否对所有时间存在，从而排除在遵循布朗流时初始不同的地标发生碰撞。我们证明，对于一般类别的度量，包括在零空间中具有无穷小刚性运动的退化度量，地标形状空间是随机完备的，从而使黎曼布朗运动在应用领域中用于建模形状随机性具有坚实基础。
英文摘要
Landmark shape spaces arise from configurations of $n$ distinct landmarks with rigid motions factored out and scale normalized. When equipped with sufficiently regular metrics inherited from right-invariant metrics on diffeomorphism groups, the shape spaces inherit geodesic completeness from geodesics of diffeomorphisms. We now answer the corresponding stochastic question if the Riemannian Brownian motion on landmark shape spaces exists for all time and hence rule out initially distinct landmarks colliding when following a Brownian flow. We show that with general classes of metrics, including degenerate metrics with infinitesimal rigid motions in their null space, landmark shape spaces are stochastically complete, thus making the use of Riemannian Brownian motion for modelling shape stochasticity in applied fields well-founded.
Comments43 pages
