Abstract
This chapter extends the discussion of stochastic differential equations and Fokker–Planck equations on Euclidean space initiated in Chapter 4 to the case of processes that evolve on a Riemannian manifold. The manifold either can be embedded in ℝn or can be an abstract manifold with Riemannian metric defined in coordinates. Section 8.1 formulates SDEs and Fokker–Planck equations in a coordinate patch. Section 8.2 formulates SDEs for an implicitly defined embedded manifold using Cartesian coordinates in the ambient space. Section 8.3 focuses on Stratonovich SDEs on manifolds. The subtleties involved in the conversion between Itô and Stratonovich formulations are explained. Section 8.4 explores entropy inequalities on manifolds. In Section 8.5 the following examples are used to illustrate the general methodology: (1) Brownian motion on the sphere and (2) the stochastic kinematic cart described in Chapter 1. Section 8.6 discusses methods for solving Fokker–Planck equations on manifolds. Exercises involving numerical implementations are provided at the end of the chapter. The main points to take away from this chapter are: SDEs and Fokker–Planck equations can be formulated for stochastic processes in any coordinate patch of a manifold in a way that is very similar to the case of Rn; Stochastic processes on embedded manifolds can also be formulated extrinsically, i.e., using an implicit description of the manifold as a system of constraint equations; In some cases Fokker–Planck equations can be solved using separation of variables; Practical examples of this theory include Brownian motion on the sphere and the kinematic cart with noise.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Applebaum, D., Kunita, H., “Lévy flows on manifolds and Lévy processes on Lie groups,” J. Math. Kyoto Univ., 33/34, pp. 1103–1123, 1993.
Atiyah, M., Bott, R., Patodi, V.K., “On the heat equation and the Index theorem,” Invent. Math., 19, pp. 279–330, 1973.
Brockett, R.W., “Lie algebras and Lie groups in control theory,” in Geometric Methods in System Theory (D.Q. Mayne and R.W. Brockett, eds.), Reidel, Dordrecht, 1973.
Brockett, R.W., “Notes on stochastic processes on manifolds,” in Systems and Control in the Twenty-First Century (C.I. Byrnes et al. eds.), Birkhäuser, Boston, 1997.
Elworthy, K.D., Stochastic Differential Equations on Manifolds, Cambridge University Press, London, 1982.
Emery, M., Stochastic Calculus in Manifolds, Springer-Verlag, Berlin, 1989.
Flügge, S., Practical Quantum Mechanics, Vol. 1 and 2, Springer-Verlag, Berlin, 1971.
Hida, T., Brownian Motion, Applications of Math. No. 11, Springer, Berlin, 1980.
Hsu, E.P., Stochastic Analysis on Manifolds, Graduate Studies in Mathematics, Vol. 38, American Mathematical Society, Providence, RI, 2002.
Ikeda, N., Watanabe, S., Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-Holland, Amsterdam, 1989.
Itô, K., “Stochastic differential equations in a differentiable manifold,” Nagoya Math. J., 1, pp. 35–47, 1950.
Itô, K., “Stochastic differential equations in a differentiable manifold (2),” Sci. Univ. Kyoto Math. Ser. A, 28, pp. 81–85, 1953.
Itô, K., McKean, H.P., Jr., Diffusion Processes and their Sample Paths, Springer, Berlin, 1996.
Kunita, H., Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, London, 1997.
Lewis, J., “Brownian motion on a submanifold of Euclidean space,” Bull. London Math. Soc., 18, pp. 616–620, 1986.
McKean, H., Jr., Singer, I.M., “Curvature and the eigenvalues of the Laplacian,” J. Diff. Geom., 1, pp. 43–69, 1967.
McKean, H.P., Jr., “Brownian motions on the 3-dimensional rotation group,” Mem. College Sci. Univ. Kyoto Ser. A, 33, 1, pp. 25–38, 1960.
McLachlan, N.W., Theory and Application of Mathieu Functions, Oxford, Clarendon Press, 1951.
Øksendal, B., Stochastic Differential Equations, An Introduction with Applications, 5th ed., Springer, Berlin, 1998.
Orsingher, E., “Stochastic motions on the 3-sphere governed by wave and heat equations,” J. Appl. Prob., 24, pp. 315–327, 1987.
Perrin, P.F., “Mouvement Brownien D'un Ellipsoide (I). Dispersion Diélectrique Pour des Molécules Ellipsoidales,” J. Phys. Radium, 7, pp. 497–511, 1934.
Perrin, P.F., “Mouvement Brownien D'un Ellipsoide (II). Rotation Libre et Dépolarisation des Fluorescences. Translation et Diffusion de Molécules Ellipsoidales,” J. Phys. Radium, 7, pp. 1–11, 1936.
Perrin, P.F., “Étude Mathématique du Mouvement Brownien de Rotation,” Ann. Sci. Éc. Norm. Supér., 45, pp. 1–51, 1928.
Pinsky, M., “Isotropic transport process on a Riemannian manifold,” TAMS, 218, pp. 353–360, 1976.
Pinsky, M., “Can you feel the shape of a manifold with Brownian motion?” in Topics in Contemporary Probability and its Applications, pp. 89–102, (J.L. Snell, ed.), CRC Press, Boca Raton, FL, 1995.
Roberts, P.H., Ursell, H.D., “Random walk on a sphere and on a Riemannian manifold,” Philos. Trans. R. Soc. London, A252, pp. 317–356, 1960.
Stroock, D.W., An Introduction to the Analysis of Paths on a Riemannian Manifold, Mathematical Surveys and Monographs, Vol. 74, American Mathematical Society, Providence, RI, 2000.
Wang, Y., Zhou, Y., Maslen, D.K., Chirikjian, G.S., “Solving the phase-noise Fokker–Planck equation using the motion-group Fourier transform,” IEEE Trans. Commun., 54, pp. 868–877, 2006.
Yau, S.-T., “On the heat kernel of a complete Riemannian manifold,” J. Math. Pures Appl., Ser. 9, 57, pp. 191–201, 1978.
Yosida, K., “Integration of Fokker–Planck's equation in a compact Riemannian space,” Ark. Mat., 1, Nr. 9, pp. 71–75, 1949.
Yosida, K., “Brownian motion on the surface of the 3-sphere,” Ann. Math. Stat., 20, pp. 292–296, 1949.
Yosida, K., “Brownian motion in a homogeneous Riemannian space,” Pacific J. Math., 2, pp. 263–296, 1952.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2009 Birkhäuser Boston
About this chapter
Cite this chapter
Chirikjian, G.S. (2009). Stochastic Processes on Manifolds. In: Stochastic Models, Information Theory, and Lie Groups, Volume 1. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4803-9_8
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4803-9_8
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4802-2
Online ISBN: 978-0-8176-4803-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)