Probability measures on infinite-dimensional Stiefel manifolds
Probability measures on infinite-dimensional Stiefel manifolds
复制标题
无限维 Stiefel 流形上的概率测度
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
A. Mennucci
中科院分区：
文献类型：
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作者：
Eleonora Bardelli;A. Mennucci
An interest in infinite-dimensional manifolds has recently appeared in Shape Theory. An example is the Stiefel manifold, that has been proposed as a model for the space of immersed curves in the plane. It may be useful to define probabilities on such manifolds. Suppose that egin{document}$H$end{document} is an infinite-dimensional separable Hilbert space. Let egin{document}$Ssubset H$end{document} be the sphere, egin{document}$pin S$end{document} . Let egin{document}$mu$end{document} be the push forward of a Gaussian measure egin{document}$gamma$end{document} from egin{document}$T_p S$end{document} onto egin{document}$S$end{document} using the exponential map. Let egin{document}$vin T_p S$end{document} be a Cameron-Martin vector for egin{document}$gamma$end{document} ; let egin{document}$R$end{document} be a rotation of egin{document}$S$end{document} in the direction egin{document}$v$end{document} , and egin{document}$
u=R_# mu$end{document} be the rotated measure. Then egin{document}$mu,
u$end{document} are mutually singular. This is counterintuitive, since the translation of a Gaussian measure in a Cameron-Martin direction produces equivalent measures. Let egin{document}$gamma$end{document} be a Gaussian measure on egin{document}$H$end{document} ; then there exists a smooth closed manifold egin{document}$Msubset H$end{document} such that the projection of egin{document}$H$end{document} to the nearest point on egin{document}$M$end{document} is not well defined for points in a set of positive egin{document}$gamma$end{document} measure. Instead it is possible to project a Gaussian measure to a Stiefel manifold to define a probability.
An interest in infinite-dimensional manifolds has recently appeared in Shape Theory. An example is the Stiefel manifold, that has been proposed as a model for the space of immersed curves in the plane. It may be useful to define probabilities on such manifolds. Suppose that egin{document}$H$end{document} is an infinite-dimensional separable Hilbert space. Let egin{document}$Ssubset H$end{document} be the sphere, egin{document}$pin S$end{document} . Let egin{document}$mu$end{document} be the push forward of a Gaussian measure egin{document}$gamma$end{document} from egin{document}$T_p S$end{document} onto egin{document}$S$end{document} using the exponential map. Let egin{document}$vin T_p S$end{document} be a Cameron-Martin vector for egin{document}$gamma$end{document} ; let egin{document}$R$end{document} be a rotation of egin{document}$S$end{document} in the direction egin{document}$v$end{document} , and egin{document}$
u=R_# mu$end{document} be the rotated measure. Then egin{document}$mu,
u$end{document} are mutually singular. This is counterintuitive, since the translation of a Gaussian measure in a Cameron-Martin direction produces equivalent measures. Let egin{document}$gamma$end{document} be a Gaussian measure on egin{document}$H$end{document} ; then there exists a smooth closed manifold egin{document}$Msubset H$end{document} such that the projection of egin{document}$H$end{document} to the nearest point on egin{document}$M$end{document} is not well defined for points in a set of positive egin{document}$gamma$end{document} measure. Instead it is possible to project a Gaussian measure to a Stiefel manifold to define a probability.