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Time Reversions of Markov Processes

Published online by Cambridge University Press:  22 January 2016

Masao Nagasawa
Masao Nagasawa*
Affiliation:
Mathematical Institute, Nagoya University
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A time reversion of a Markov process was discussed by Kolmogoroff for Markov chains in 1936 [6] and for a diffusion in 1937 [7l He described it as a process having an adjoint transition probability. Although his treatment is purely analytical, in his case if the process xt has an invariant distribution, the reversed process zt = x-t is the process with the adjoint transition probability. In this discussion, however, it is very restrictive that the initial distribution of the process must be an invariant measure.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 24 , June 1964 , pp. 177 - 204
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1964

References

[1] Dynkin, E. B., Markov processes. Moskow (1963), (Russian).Google Scholar
[2] Hunt, G. A., Markoff chains and Martin boundaries. III. J. Math. 4 (1960), 316–340.Google Scholar
[3] Hunt, G. A., Markoff processes and potentials, III. J. Math. 2 (1958), 151–213.Google Scholar
[4] Ikeda, N., Sato, K., Tanaka, H. and Ueno, T., Boundary problems in multi-dimensisnal diffusion process, I, II. Seminar on Prob. 5 (1960), 6 (1961), (Japanese).Google Scholar
[5] Ikeda, N., Nagasawa, M. and Sato, K., A time reversion of Markov processes with killing. Kōdai Math. Sem. Rep. 16 (1964), 88–97.Google Scholar
[6] Kolmogoroff, A., Zur Theorie der Markoffschen Ketten. Math. Ann. 112 (1936), 155–160.CrossRefGoogle Scholar
[7] Kolmogoroff, A., Zur Unkehrbarkeit der statistischen Naturgesetze, Math. Ann. 113 (1937), 766–772.CrossRefGoogle Scholar
[8] Kunita, H. and Watanabe, T., Note on transformations of Markov processes connected with multiplicative functionals. Mem. Sci Fac.. Kyushu Univ. Series A. Math. 17 (1963), 181–191.Google Scholar
[9] McKean, H. P. Jr. and Tanaka, H., Additive functionals of the Brownian path. Mem. Coll. Sci. Univ. Kyoto, Ser. A 33 (1961), 476–506.Google Scholar
[10] Meyer, P. A., Fonctionnelles multiplicatives et additives de Markov. Ann. Inst. Fourier, 12 (1962), 125–230.Google Scholar
[11] Motoo, M., Additive functionals of Markov processes. Seminar on Prob. 15 (1963), 1–123, (Japanese).Google Scholar
[12] Nagasawa, M., The adjoint process of a diffusion with reflecting barrier, Kōdai Math. Sem. Rep. 13 (1961), 235–248.Google Scholar
[13] Nagasawa, M., and Sato, K., Remarks to “The adjoint process of a diffusion with reflecting barrier” Kōdai Math. Sem. Rep. 14 (1962), 119–122.Google Scholar
[14] Nagasawa, M. and Sato, K., Some theorems on time change and killing of Markov processes. Kōdai Math. Sem. Rep. 15 (1963), 195–219.Google Scholar
[15] Nelson, E., The adjoint Markov process. Duke Math. J. 25 (1958), 671–690.CrossRefGoogle Scholar
[16] Sur, M. G., Continuous additive functionals of Markov processes and excessive functions, Dokl. Acad. Nauk. SSSR 137 (1961), 800–803, (Russian).Google Scholar
[17] Volkonskii, V. A., Additive functionals of Markov processes, Trudy Mosk. Mat. Ob. 9 (1960), 143–189, (Russian).Google Scholar