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Time-reversible diffusions

Published online by Cambridge University Press:  01 July 2016

John Kent
John Kent*
Affiliation:
University of Leeds
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Abstract

Symmetric diffusions on manifolds with boundary are studied. Symmetric diffusions are nicer to work with than non-symmetric diffusions because (1) it is easier to tell if an equilibrium density exists, and (2) it is easier to find the equilibrium density when it does exist. If an equilibrium density exists, a symmetric diffusion is time reversible. On the line essentially all diffusions are symmetric.

Using symmetric diffusions, it is shown that a large family of densities can be realized as equilibrium densities of time-reversible diffusions. Some examples are given.

Keywords

DIFFUSIONTIME REVERSIBILITYSYMMETRIC DIFFUSIONCHARACTERISTIC DIFFUSION
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 4 , December 1978 , pp. 819 - 835
Copyright
Copyright © Applied Probability Trust 1978 

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References

Boothby, W. M. (1975) An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, New York.Google Scholar
Hicks, N. J. (1965) Notes on Differential Geometry. Van Nostrand, New York.Google Scholar
Ito, K. and McKean, H. P. Jr. (1965) Diffusion Processes and their Sample Paths. Springer-Verlag, Berlin.Google Scholar
Ito, S. (1957) The fundamental solution of the parabolic differential equation in a differentiable manifold, II. Osaka Math. J. 6, 167–185.Google Scholar
Ito, S. (1957) Fundamental solutions of parabolic differential equations and boundary value problems. Japan. J. Maths 27, 55–102.Google Scholar
Kendall, D. G. (1975) Some problems in mathematical genealogy. In Perspectives in Probability and Statistics, ed. Gani, J.. Distributed by Academic Press, London for the Applied Probability Trust, Sheffield, 325–345.Google Scholar
Kent, J. T. (1975) Discussion following Mardia (1975). J.R.Statist. Soc. B37, 377.Google Scholar
Khas'minskii, R. Z. (1960) Ergodic properties of recurrent diffusion processes and stabilization of the Cauchy problem for parabolic equations. Theory Prob. Appl. 5, 196–214.Google Scholar
Kolmogorov, A. (1937) Zur Umkehrbarkeit der statistischen Naturgesetze. Math. Annalen 113, 766–772.Google Scholar
Mandl, P. (1968) One-Dimensional Markov Processes. Springer-Verlag, Berlin.Google Scholar
Mardia, K. V. (1972) Statistics of Directional Data. Academic Press, London.Google Scholar
Mardia, K. V. (1975) Statistics of directional data (with discussion). J.R.Statist. Soc. B37, 349–393.Google Scholar
Muller, C. (1966) Spherical Harmonics. Springer-Verlag, Berlin.Google Scholar
Ripley, B. D. (1977) Modelling spatial patterns (with discussion). J.R.Statist. Soc. B39, 172–212.Google Scholar
Silverstein, M. L. (1974) Symmetric Markov Processes. Lecture Notes in Mathematics 426, Springer-Verlag, Berlin.Google Scholar
Silverstein, M. L. (1976) Boundary Theory for Symmetric Markov Processes. Lecture Notes in Mathematics 516, Springer-Verlag, Berlin.Google Scholar
Whittle, P. (1975) Reversibility and acyclicity. In Perspectives in Probability and Statistics, ed. Gani, J.. Distributed by Academic Press, London, for the Applied Probability Trust, Sheffield, 217–224.Google Scholar