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Brownian motions on shape and size-and-shape spaces

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Huiling Le
Huiling Le*
Affiliation:
University of Nottingham
*
∗ Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
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Abstract

The diffusions on the shape and size-and-shape spaces induced by brownian motions on the pre-size-and-shape spaces have been investigated in several papers (cf.). We here address the dual problem: the character of the diffusions on the pre-shape and pre-size-and-shape spaces which induce brownian motions on the shape and size-and-shape spaces. In particular we show that the shape and size-and-shape spaces for k labelled points in ℝm are stochastically complete if k > m and obtain the heat kernels of certain diffusions which induce brownian motions on the size-and-shape spaces.

Keywords

HEAT KERNELSEXPLOSION TIME

MSC classification

Secondary: 60J65: Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 101 - 113
Copyright
Copyright © Applied Probability Trust 1994 

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References

[1] Carne, T. K. (1990) The geometry of shape spaces. Proc. London Math. Soc. 61, 407–432.Google Scholar
[2] Bru, ?. F. (1989) Diffusions of perturbed principal component analysis. J. Multivariate Analysis 29, 127–136.CrossRefGoogle Scholar
[3] Elworthy, K. D. (1982) Stochastic Differential Equations on Manifolds. Cambridge University Press.Google Scholar
[4] Helgason, S. (1984) Groups and Geometric Analysis. Academic Press, New York.Google Scholar
[5] Kendall, D. G. (1984) Shape manifolds, procrustean metrics, and complex projective space. Bull. London Math. Soc. 16, 81–121.Google Scholar
[6] Kendall, W. S. (1990) The diffusion of euclidean shape. In Disorder in Physical Systems, ed. Grimmett, G. and Welsh, D., pp. 203–217, Cambridge University Press.Google Scholar
[7] Le, H. and Kendall, D. G. (1992) The riemannian structure of euclidean shape spaces: a novel environment for statistics. Ann. Statist. (to appear).Google Scholar
[8] Norris, J. R., Rogers, L. C. G. and Williams, D. (1986) Brownian motions of ellipsoids. Trans. Amer. Math. Soc. 294, 757–765.Google Scholar
[9] Pauwels, E. J. (1990) Riemannian submersions of brownian motions. Stoch. Stoch. Rep. 29, 425–436.Google Scholar
[10] Revuz, D. and Yor, M. (1991) Continuous Martingales and Brownian Motion. Springer-Verlag, New York.Google Scholar
[11] Rogers, L. C. G. and Williams, D. (1987) Diffusions, Markov Processes, and Martingales. Wiley, New York.Google Scholar
[12] Yor, M. (1980) Loi de l'indice du lacet brownien et distribution de Hartman-Watson. Z. Wahrscheinlichkeitsth. 53, 71–95.Google Scholar