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Comparison Theorems on Regular Points for Multidimensional Markov Processes of Transient Type1)

Published online by Cambridge University Press:  22 January 2016

Mamoru Kanda
Mamoru Kanda*
Affiliation:
Hiroshima University
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The study of regular points for the Dirichlet problem has a long history. The probabilistic approach to regular points is originated by Doob [2] and [3] for Brownian motion and the heat process. The extension to general Markov processes is discussed in Dynkin [4] and [5]. They also clarified the relation between the fine topology and regular points.

Regular points are by definition reflected in the behaviour of sample paths of Markov processes. Further the inclusion relation of collections of regular points for open sets determines the strength and the weakness of fine topologies between two processes. Hence it is meaningful to compare the collections of regular points for compact or open sets between two Markov processes apart from the Dirichlet problem.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 44 , December 1971 , pp. 165 - 214
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

Footnotes

1)

This research was supported in part by Yukawa Foundation.

References

Blbilography

[1] Blumenthal, R.M. and Getoor, R.K., Markov processes and potential theory. Academic Press, New York and London, 1968.Google Scholar
[2] Doob, J.L., Semimartingales and subharmonic functions. Trans. Am. Math. Soc, 77, 86–121 (1954).Google Scholar
[3] Doob, J.L., A probability approach to the heat equation. Trans. Am. Math. Soc, 80, 216–280 (1955).Google Scholar
[4] Dynkin, E.B., Intrinsic topology and excessive functions connected with a Markov process. Dokl. Akad. Nauk SSSR 127 17–19, (1959).Google Scholar
[5] Dynkin, E.B., Markov processes.” Moscow, 1963. English translation (in two volumes): Springer, Berlin, 1965.Google Scholar
[6] Feller, W., An introduction to Probability Theory and its Applications,” Vol. 11 Wiley, New York, 1966.Google Scholar
[7] Girbarg, D. and Serrin, J., On isolated singularities of solutions of second order elliptic differential equations, J. Analyse Math. 4 309–340 (1954–1955).CrossRefGoogle Scholar
[8] Hörmander, L., Pseudo-differential operators, Comm. Pure Appl. Math. 18 501–517 (1965).CrossRefGoogle Scholar
[9] Hunt, G.A., Markoff processes and potentials I., Illinois J. Math. 144–93 (1957).Google Scholar
[10] Hunt, G.A. Markoff processes and potentials. Ill, Illinois J. Math. 2 151–213 (1958).CrossRefGoogle Scholar
[11] Ikeda, N. and Watanabe, S., On some relations between the harmonic measure and the Levy measure for a certain class of Markov processes. J. Math., Kyoto Univ., 1–1 79–95 (1962).Google Scholar
[12] Il’in, A.M., Kalashnikov, A.S. and Oleinik, O.A., Second order linear equations of parabolic type. Uspekhi Math. Nauk 17 3–146 (1962).Google Scholar
[13] Kanda, M., Regular points and Green functions in Markov processes. J. Math. Soc. Japan 19 46–69 (1967).Google Scholar
[14] Kanda, M., On the singularity of Green functions in Markov Processes. Nagoya Math. J., 33 21–52 (1968).Google Scholar
[15] Kanda, M., On the singularity of Green functions in Markov processes II. Nagoya Math. J., 37 209–217 (1970).Google Scholar
[16] Kohn, J.J. and Nirenberg, L., An algebra of pseudo-differential operators. Comm. Pure Appl. Math., 18 269–305 (1965).Google Scholar
[17] Krylov, N.V., On quasi-diffusion processes. Theory of Probability and its Applications 11 424–443 (1966).CrossRefGoogle Scholar
[18] Krylov, N.V., On the green function for the Dirichlet problem. Uspekhi Math. Nauk 22 116–118 (1967).Google Scholar
[19] Krylov, N.V., The first boundary value problem for elliptic equations of second order. Differential Equations 3 315–325 (1967).Google Scholar
[20] Landis, E.M., S-capacity and its applications to the study of solutions of second order elliptic differential equations with discontinuous coefficients. Math. Sbornik 5 177–204 (1968).Google Scholar
[21] Landkof, N.S., “Foundamentals of modern potential theory.” Moscow, 1966.Google Scholar
[22] Motoo, M., The sweeping-out of additive functionals and processes on the boundary. Ann. Inst. Statist. Math., 16 317–345 (1964).Google Scholar
[23] Palais, R.S., “Seminar on the Atiyah-Singer index theorem.” Annals of Mathematics Studies 57 Princeton, New Jersey, 1965.Google Scholar
[24] Schwartz, L. “Theorie des distributions.” Hermann Sc Cie Paris, 1950.Google Scholar
[25] Skorokhad, A.V., Asymptotic formulas for stable distribution laws. Dokl. Akad. Nauk. SSSR 98 731–734 (1954).Google Scholar
[26] Skorokhod, A.V., “Studies in the Theory of Random Processes.” Kiev 1961. English translation Addison-Wesley, 1965.Google Scholar
[27] Stroock, D.W. and Varadhan, S.R.S., Diffusion processes with continuous coefficients I, II. Comm. Pure Appl. Math. 21 and 22 479–530 (1969).Google Scholar
[28] Tanaka, H., Existence of diffusions with continuous coefficients. Mem. Fac. Sci. Kyushu Univ. 18 89–103 (1964).Google Scholar
[29] Taylor, S.J., On the connection between Hausdorff measures and generalized capacity. Proc. Cambridge Philos. Soc. 57 524–531 (1961).Google Scholar