We study Brownian motion on the space of distinct landmarks in , considered as a homogeneous space with a Riemannian metric inherited from a right-invariant metric on the diffeomorphism group. As of yet, there is no proof of long-time existence of this process, despite its fundamental importance in statistical shape analysis, where it is used to model stochastic shape evolutions. We make some first progress in this direction by providing a full classification of long-time existence for configurations of exactly two landmarks, governed by a radial kernel. For low-order Sobolev kernels, we show that the landmarks collide with positive probability in finite time, whilst for higher-order Sobolev and Gaussian kernels, the landmark Brownian motion exists for all times. We illustrate our theoretical results by numerical simulations.

中文翻译：

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两个地标构型上布朗运动的长期存在性

我们研究了不同地标空间上的布朗运动，被视为具有从微分同胚群上的右不变度量继承的黎曼度量的齐次空间。到目前为止，还没有证据证明这一过程长期存在，尽管它在统计形状分析中具有根本重要性，用于模拟随机形状演化。我们在这个方向上取得了一些初步进展，为由径向核控制的恰好两个地标的配置提供了长期存在的完整分类。对于低阶 Sobolev 核，我们表明地标在有限时间内以正概率发生碰撞，而对于高阶 Sobolev 核和高斯核，地标布朗运动始终存在。我们通过数值模拟来说明我们的理论结果。