The resolution of an irregularity of boundary points in the boundary problem for Markov processes

Yasunori OKABE

doi:10.2969/jmsj/02210047

Yasunori OKABE

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1970 Volume 22 Issue 1 Pages 47-104

DOI https://doi.org/10.2969/jmsj/02210047

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Published: 1970 Received: June 09, 1969 Available on J-STAGE: September 29, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: September 29, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) M. Brelot, Sur le principe des singularites positives et la topologie de R.S. Martin, Ann. Univ. Grenoble, Section Math. Phys., 23 (1947/48), 113-138.
2) M. Brelot, On Martin boundary, April 30, May 1, 2, 4 (1962), at Hiroshima University (Lecture Note).
3) C. Constantinescu and A. Cornea, Ideale Rander Riemannscher Flachen, Springer, 1963.
4) C. Constantinescu and A. Cornea, Compactifications of harmonic spaces, Nagoya Math. J., 30 (1965), 1-57.
5) J.L. Doob, Compactification of the discrete state space of a Markov process, Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 10 (1968), 236-251.
6) E.B. Dynkin, Foundations of the theory of Markov processes, English translation; Pergamon Press, 1960.
7) E.B. Dynkin, General boundary conditions for denumerable Markov processes, Teor. Veroyatnost. i Primenen., 12 (1967), 187-221 (Russian).
8) E.B. Dynkin, On extensions of a Markov process, Teor. Veroyatnost. i Primenen., 13 (1968), 708-713 (Russian).
9) K. Ito, Probability Theory, Iwanami Schoten, Tokyo, 1953 (Japanese).
10) Y, Kawada and Y. Mimura, General concepts of modern mathematics, II, Iwanami Schoten, Tokyo, 1965 (Japanese).
11) H. Kunita and T. Watanabe, Some theorems concerning resolvents over locally compact spaces, Proc. 5th Berkeley Symp. Math. Statist. and Prob. II, Part 2 (1967), 131-167.
12) H. Kunita, K. Sato, M. Fukushima and M. Motoo, Diffusion Process and Markov Processes on the boundary, Seminar on Probability, 22 (1965) (Japanese).
13) P.A. Meyer, Compactifications associées à une resolvente, Seminaire de ProbabilitésII in Lecture Note in Math., 51, 175-199, Berlin-Heidelberg-New York; Springer 1968.
14) M. Motoo, Representation of a certain class of excessive functions and a generator of Markov processes, Sci. Papers Coll. Gen. Ed. Univ. Tokyo, 12 (1962), 143-159.
15) M. Motoo, Additive functional of Markov processes, Seminar on Probability, 15 (1963) (Japanese).
16) M. Motoo, The sweeping-out of additive functionals and processes on the boundary, Ann. Inst. Statist. Math., 16 (1964), 317-345.
17) M. Motoo, Application of additive functionals to the boundary problem of Markov processes (Lévy system of U-processes), Proc. 5th Berkeley Symp. Math. Statist. and Prob. II, Part 2 (1967), 75-110.
18) D. Ray, Resolvents, transition functions, and strongly Markovian processes, Ann. of Math., II, Ser. 70 (1959), 43-72.
19) K. Sato, A decomposition of Markov processes, J. Math. Soc. Japan, 17 (1965), 219-243.
20) T. Ueno, A survey on the Markov processes on the boundary of multi-dimensional diffusion, Proc. 5th Berkeley Symp. Math. Statist. and Prob. II, Part 2 (1967), 111-130.
21) S. Watanabe, On discontinuous additive functionals and Lévy measures of a Markov process, Japan J. Math., 34 (1964), 53-70.
22) K. Yosida, Functional Analysis, Springer, 1966.

Right : [1] M. Brelot, Sur le principe des singularites positives et la topologie de R. S. Martin, Ann. Univ. Grenoble, Section Math. Phys., 23 (1947/48), 113-138.
[2] M. Brelot, On Martin boundary, April 30, May 1, 2, 4 (1962), at Hiroshima University (Lecture Note).
[3] C. Constantinescu and A. Cornea, Ideale Rander Riemannscher Flachen, Springer, 1963.
[4] C. Constantinescu and A. Cornea, Compactifications of harmonic spaces, Nagoya Math. J., 30 (1965), 1-57.
[5] J. L. Doob, Compactification of the discrete state space of a Markov process, Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 10 (1968), 236-251.
[6] E. B. Dynkin, Foundations of the theory of Markov processes, English translation; Pergamon Press, 1960.
[7] E. B. Dynkin, General boundary conditions for denumerable Markov processes, Teor. Veroyatnost. i Primenen., 12 (1967), 187-221 (Russian).
[8] E. B. Dynkin, On extensions of a Markov process, Teor. Veroyatnost. i Primenen., 13 (1968), 708-713 (Russian).
[9] K. Ito, Probability Theory, Iwanami Schoten, Tokyo, 1953 (Japanese).
[10] Y, Kawada and Y. Mimura, General concepts of modern mathematics, II, Iwanami Schoten, Tokyo, 1965 (Japanese).
[11] H. Kunita and T. Watanabe, Some theorems concerning resolvents over locally compact spaces, Proc. 5th Berkeley Symp. Math. Statist. and Prob. II, Part 2 (1967), 131-167.
[12] H. Kunita, K. Sato, M. Fukushima and M. Motoo, Diffusion Process and Markov Processes on the boundary, Seminar on Probability, 22 (1965) (Japanese).
[13] P. A. Meyer, Compactifications associées à une resolvente, Seminaire de Probabilités II in Lecture Note in Math., 51, 175-199, Berlin-Heidelberg-New York; Springer 1968.
[14] M. Motoo, Representation of a certain class of excessive functions and a generator of Markov processes, Sci. Papers Coll. Gen. Ed. Univ. Tokyo, 12 (1962), 143-159.
[15] M. Motoo, Additive functional of Markov processes, Seminar on Probability, 15 (1963) (Japanese).
[16] M. Motoo, The sweeping-out of additive functionals and processes on the boundary, Ann. Inst. Statist. Math., 16 (1964), 317-345.
[17] M. Motoo, Application of additive functionals to the boundary problem of Markov processes (Lévy system of U-processes), Proc. 5th Berkeley Symp. Math. Statist. and Prob. II, Part 2 (1967), 75-110.
[18] D. Ray, Resolvents, transition functions, and strongly Markovian processes, Ann. of Math., II, Ser. 70 (1959), 43-72.
[19] K. Sato, A decomposition of Markov processes, J. Math. Soc. Japan, 17 (1965), 219-243.
[20] T. Ueno, A survey on the Markov processes on the boundary of multi-dimensional diffusion, Proc. 5th Berkeley Symp. Math. Statist. and Prob. II, Part 2 (1967), 111-130.
[21] S. Watanabe, On discontinuous additive functionals and Lévy measures of a Markov process, Japan J. Math., 34 (1964), 53-70.
[22] K. Yosida, Functional Analysis, Springer, 1966.

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