Semi-classical analysis of a random walk on a manifold
Gilles Lebeau and Laurent Michel
Source: Ann. Probab. Volume 38, Number 1 (2010), 277-315.
Abstract
We prove a sharp rate of convergence to stationarity for a natural random walk on a compact Riemannian manifold (M, g). The proof includes a detailed study of the spectral theory of the associated operator.
Primary Subjects: 58J65, 60J10, 35S05
Keywords: Random walk; Metropolis; pseudo-differential calculus
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1264433999
Digital Object Identifier: doi:10.1214/09-AOP483
Zentralblatt MATH identifier:
05678680
References
[1] Bhattacharya, R. and Patrangenaru, V. (2002). Nonparametic estimation of location and dispersion on Riemannian manifolds. J. Statist. Plann. Inference 108 23–35.
[2] Bhattacharya, R. and Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I. Ann. Statist. 31 1–29.
[3] Diaconis, P., Holmes, S. and Shahshahani, M. (2007). Sampling from a manifold. Preprint, Dept. Statistics, Stanford Univ.
[4] Diaconis, P. and Lebeau, G. (2009). Micro-local analysis for the Metropolis algorithm. Math. Z. 262 411–447.
[5] Diaconis, P., Lebeau, G. and Michel, L. (2008). Geometric analysis for the Metropolis algorithm on Lipschitz domains. Nice University. Preprint. Available at http://math.unice.fr/~lmichel/recherche.html.
[6] Diaconis, P. and Saloff-Coste, L. (1998). What do we know about the Metropolis algorithm? J. Comput. System Sci. 57 20–36.
[7] Dimassi, M. and Sjöstrand, J. (1999). Spectral Asymptotics in the Semi-Classical Limit. Lecture Note Series 268. Cambridge Univ. Press, London.
[8] Émery, M. (1989). Stochastic Calculus in Manifolds. Universitext. Springer, Berlin.
[9] Fukushima, M., Ōshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics 19. de Gruyter, Berlin.
[10] Giné, M. E. (1975). Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms. Ann. Statist. 3 1243–1266.
[11] Hsu, E. P. (2002). Stochastic Analysis on Manifolds. Graduate Studies in Mathematics 38. Amer. Math. Soc., Providence, RI.
[12] Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics 113. Springer, New York.
[13] Martinez, A. (2002). An Introduction to Semiclassical and Microlocal Analysis. Springer, New York.
[14] Sogge, C. D. (1988). Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds. J. Funct. Anal. 77 123–138.
[15] Taylor, M. E. (1997). Partial Differential Equations. III. Applied Mathematical Sciences 117. Springer, New York.
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