In statistics on manifolds, the notion of the mean of a probability distribution becomes more involved than in a linear space. Several location statistics have been proposed, which reduce to the ordinary mean in Euclidean space. A relatively new family of contenders in this field are Diffusion Means, which are a one parameter family of location statistics modeled as initial points of isotropic diffusion with the diffusion time as parameter. It is natural to consider limit cases of the diffusion time parameter and it turns out that for short times the diffusion mean set approaches the intrinsic mean set. For long diffusion times, the limit is less obvious but for spheres of arbitrary dimension the diffusion mean set has been shown to converge to the extrinsic mean set. Here, we extend this result to the real projective spaces in their unique smooth isometric embedding into a linear space. We conjecture that the long time limit is always given by the extrinsic mean in the isometric embedding for connected compact symmetric spaces with unique isometric embedding.

翻译：在流形统计学中，概率分布均值的概念比线性空间中的情况更为复杂。已有多种位置统计量被提出，它们在欧几里得空间中可退化为普通均值。该领域中一个相对较新的竞争者是扩散均值族，它是一个单参数族的位置统计量，被建模为各向同性扩散的初始点，其中扩散时间作为参数。自然需要考虑扩散时间参数的极限情况，结果表明在短时间下，扩散均值集会趋近于本征均值集。对于长时间扩散，其极限较不明显，但对于任意维度的球面，已证明扩散均值集会收敛于外征均值集。本文将此结果推广到实射影空间在其唯一光滑等距嵌入线性空间的情形。我们推测，对于具有唯一等距嵌入的连通紧对称空间，其长时间极限总是由等距嵌入中的外征均值给出。