On boundary conditions for multi-dimensional Brownian motions with symmetric resolvent densities

Masatoshi FUKUSHIMA

doi:10.2969/jmsj/02110058

Masatoshi FUKUSHIMA

著者情報

ジャーナルフリー

1969 年 21 巻 1 号 p. 58-93

DOI https://doi.org/10.2969/jmsj/02110058

詳細

Published: 1969 Received: 1968/01/07 Available on J-STAGE: 2006/09/29 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/09/29 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) A. Beurling and J. Deny, Espaces de Dirichlet, Le cas elementaire, Acta Math., 99 (1958), 203-224.
2) A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 208-215.
3) M. Brelot, Etude et extensions du principe de Dirichlet, Ann. Inst. Fourier, 5 (1953-4), 371-419.
4) J. Deny, Principe complet du maximum et contractions, Ann. Inst. Fourier, 15 (1965), 259-272.
5) J. Deny et J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier, 5 (1953-4), 305-370.
6) J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France, 85 (1957), 431-458.
7) J. L. Doob, Boundary properties of functions with finite Dirichlet integrals, Ann. Inst. Fourier, 12 (1952), 573-621.
8) E. B. Dynkin, General boundary conditions for denumerable Markov processes (in Russian), Teor. Veroyatnost. i Primenen., 12 (1967), 222-257.
9) E. B. Dynkin, General lateral conditions for some diffusion processes, Proc. 5th Berkeley Sympos. Math. Statist. and Prob.
10) J. Elliott, Dirichlet spaces associated with integro-differential operators, I, II,Illinois J. Math., 9 (1965), 87-98; 10 (1966), 66-89.
11) W. Feller, On boundaries and lateral conditions for the Kolmogorov differential equations, Ann. Math., 65 (1957), 527-570.
12) W. Feller, The parabolic differential equations and the associated semigroups of transformations, Ann, of Math., 55 (1952), 468-519.
13) M. Fukushima, On Feller's kernel and the Dirichlet norm, Nagoya Math. J., 24 (1964), 167-175.
14) M. Fukushima, Resolvent kernels on a Martin spaces, Proc. Japan Acad., 41 (1964), 167-175.
15) M. Fukushima, A construction of reflecting barrier Brownian motions for bounded domains, Osaka J. Math., 4 (1967), 183-215.
16) K. Ito and H. P. Mckean Jr., Diffusion processes and their sample paths, Springer, 1965.
17) H. Kunita, Applications of Martin boundaries to instantaneous return Markov processes over a denumerable space, J. Math. Soc. Japan, 14 (1962), 66-100.
18) R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc., 49 (1941), 137-172.
19) L. Naim, Sur le rôle de la frontiere de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier, 7 (1957), 183-281.
20) J. Neveu, Une generalisation des processes à accroissements positifs independents, Abh. Math. Sem. Univ. Hamburg, 25 (1961), 36-61.
21) T. Shiga and T. Watanabe, On Markov chains similar to the reflecting barrier Brownian motion, Osaka J. Math., 5 (1968), 1-33.
22) A. V. Skorohod, Boundary conditions for certain Markov processes (in Russian), Teor. Veroyatnost. i Primenen., 9 (1964), 644-654.
23) A. D. Wentzell, On boundary conditions for multidimensional diffusion processes (in Russian), Teor. Veryatnost. i Primenen., 4 (1959), 172-185.
24) T. Watanabe, to appear.

Right : [1] A. Beurling and J. Deny, Espaces de Dirichlet, Le cas elementaire, Acta Math., 99 (1958), 203-224.
[2] A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U. S. A., 45 (1959), 208-215.
[3] M. Brelot, Etude et extensions du principe de Dirichlet, Ann. Inst. Fourier, 5 (1953-4), 371-419.
[4] J. Deny, Principe complet du maximum et contractions, Ann. Inst. Fourier, 15 (1965), 259-272.
[5] J. Deny et J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier, 5 (1953-4), 305-370.
[6] J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France, 85 (1957), 431-458.
[7] J. L. Doob, Boundary properties of functions with finite Dirichlet integrals, Ann. Inst. Fourier, 12 (1962), 573-621.
[8] E. B. Dynkin, General boundary conditions for denumerable Markov processes (in Russian), Teor. Veroyatnost. i Primenen., 12 (1967), 222-257.
[9] E. B. Dynkin, General lateral conditions for some diffusion processes, Proc. 5th Berkeley Sympos. Math. Statist. and Prob.
[10] J. Elliott, Dirichlet spaces associated with integro-differential operators, I, II,Illinois J. Math., 9 (1965), 87-98; 10 (1966), 66-89.
[11] W. Feller, On boundaries and lateral conditions for the Kolmogorov differential equations, Ann. Math., 65 (1957), 527-570.
[12] W. Feller, The parabolic differential equations and the associated semigroups of transformations, Ann. of Math., 55 (1952), 468-519.
[13] M. Fukushima, On Feller's kernel and the Dirichlet norm, Nagoya Math. J., 24 (1964), 167-175.
[14] M. Fukushima, Resolvent kernels on a Martin spaces, Proc. Japan Acad., 41 (1964), 167-175.
[15] M. Fukushima, A construction of reflecting barrier Brownian motions for bounded domains, Osaka J. Math., 4 (1967), 183-215.
[16] K. Ito and H. P. Mckean Jr., Diffusion processes and their sample paths, Springer, 1965.
[17] H. Kunita, Applications of Martin boundaries to instantaneous return Markov processes over a denumerable space, J. Math. Soc. Japan, 14 (1962), 66-100.
[18] R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc., 49 (1941), 137-172.
[19] L. Naim, Sur le rôle de la frontiere de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier, 7 (1957), 183-281.
[20] J. Neveu, Une generalisation des processus à accroissements positifs independents, Abh. Math. Sem. Univ. Hamburg, 25 (1961), 36-61.
[21] T. Shiga and T. Watanabe, On Markov chains similar to the reflecting barrier Brownian motion, Osaka J. Math., 5 (1968), 1-33.
[22] A. V. Skorohod, Boundary conditions for certain Markov processes (in Russian), Teor. Veroyatnost. i Primenen., 9 (1964), 644-654.
[23] A. D. Wentzell, On boundary conditions for multidimensional diffusion processes (in Russian), Teor. Veryatnost. i Primenen., 4 (1959), 172-185.
[24] T. Watanabe, to appear.

訂正日: 2006/09/29 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

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