Published: 1988 Received: February 23, 1987Available on J-STAGE: October 20, 2006Accepted: -
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Date of correction: October 20, 2006Reason for correction: -Correction: CITATIONDetails: Wrong : 1) W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519. 2) W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2 (2nd ed.), Wiley, New York, 1970. 3) K. Ito and H. P. McKean, Jr., Diffusion Processes and their Sample Paths, Springer, 1965. 4) I. S. Kac, Integral characteristics of the growth of spectral functions for generalized second order boundary problems with conditions at a regular end, IZV. Akad. Nauk. SSSR Ser. Mat., 38 (1971), 154-184 (in Russian), Math. USSR-Izv., 5 (1971), 161-191 (Engl. transl.). 5) I. S. Kac, Generalization of an asymptotic formula of V. A. Marcenko for spectral functions of a second order boundary value problem, Izv. Akad. Nauk. SSSR Ser. Mat., 37 (1973), 422-436 (in Russian), Math. USSR-Izv., 7 (1973), 422-436 (Engl. transl.). 6) I. S. Kac and M. G. Krein, On the spectral functions of the string, Amer. Math. Transl., Ser. 2, 103 (1974), 19-102. 7) Y. Kasahara, Spectral theory of generalized second order differential operators and its applications to Markov processes, Japan. J. Math., 1 (1975) , 67-84. 8) Y. Kasahara, S. Kotani and H. Watanabe, On the Green function of 1-dimensional diffusion processes, Publ. RIMS, Kyoto Univ., 16 (1980), 175-188. 9) S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, Functional Analysis in Markov Processes (M. Fukushima, ed.), Lecture Notes in Math., 923, Springer, 1982, pp. 235-259. 10) M. G. Krein, On some cases of the effective determination of the densities of a nonhomogeneous string from its spectral function, Dokl. Acad. Nauk. SSSR, 93(1953), 617-620. 11) H. P. McKean, Jr., Elementary solutions for certain parabolic differential equations, Trans. Amer. Math. Soc., 82 (1956), 519-548. 12) N. Minami, Y. Ogura and M. Tomisaki, Asymptotic behavior of elementary solutions of one-dimensional generalized diffusion equations, Ann. Prob., 13 (1985), 698-715. 13) Y. Ogura, One-dimensional bi-generalized diffusion processes (preprint). 14) Y. Ogura and M. Tomisaki, Asymptotic behaviors of moments for one-dimensional diffusion processes, (BiBoS preprint series). 15) Y. Okabe, On long time tails of correlation functions for KMO-Langevin equations, Probability Theory and Mathematical Statistics (S. Watanabe and Yu. V. Prokhorov, ed.), Lecture Notes in Math., 1299, Springer, 1988, pp. 391-397. 16) E. Seneta, Regular Varying Functions, Lecture Notes in Math., 508, Springer, 1976. 17) M. Tomisaki, Power order decay of elementary solutions of generalized diffusion equations, Probability Theory and Mathematical Statistics (S. Watanabe and Yu. V. Prokhorov, ed.), Lecture Notes in Math., 1299, Springer, 1988, pp. 511-523. 18) S. Watanabe, On time inversion of one-dimensional diffusion processes, Z. Wahrsch. verw. Geb., 31 (1975), 115-124. 19) K. Yoshida, Lectures on Differential and Integral Equations, Interscience, New York, 1960.
Right : [1] W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519. [2] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2 (2nd ed.), Wiley, New York, 1970. [3] K. Itô and H. P. McKean, Jr., Diffusion Processes and their Sample Paths, Springer, 1965. [4] I. S. Kac, Integral characteristics of the growth of spectral functions for generalized second order boundary problems with conditions at a regular end, Izv. Akad. Nauk. SSSR Ser. Mat., 38 (1971), 154-184 (in Russian), Math. USSR-Izv., 5 (1971), 161-191 (Engl. transl.). [5] I. S. Kac, Generalization of an asymptotic formula of V. A. Marcenko for spectral functions of a second order boundary value problem, Izv. Akad. Nauk. SSSR Ser. Mat., 37 (1973), 422-436 (in Russian), Math. USSR-Izv., 7 (1973), 422-436 (Engl. transl.). [6] I. S. Kac and M. G. Krein, On the spectral functions of the string, Amer. Math. Transl., Ser. 2, 103 (1974), 19-102. [7] Y. Kasahara, Spectral theory of generalized second order differential operators and its applications to Markov processes, Japan. J. Math., 1 (1975), 67-84. [8] Y. Kasahara, S. Kotani and H. Watanabe, On the Green function of 1-dimensional diffusion processes, Publ. RIMS, Kyoto Univ., 16 (1980), 175-188. [9] S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, Functional Analysis in Markov Processes (M. Fukushima, ed.), Lecture Notes in Math., 923, Springer, 1982, pp. 235-259. [10] M. G. Krein, On some cases of the effective determination of the densities of a nonhomogeneous string from its spectral function, Dokl. Acad. Nauk. SSSR, 93(1953), 617-620. [11] H. P. McKean, Jr., Elementary solutions for certain parabolic differential equations, Trans. Amer. Math. Soc., 82 (1956), 519-548. [12] N. Minami, Y. Ogura and M. Tomisaki, Asymptotic behavior of elementary solutions of one-dimensional generalized diffusion equations, Ann. Prob., 13 (1985), 698-715. [13] Y. Ogura, One-dimensional bi-generalized diffusion processes (preprint). [14] Y. Ogura and M. Tomisaki, Asymptotic behaviors of moments for one-dimensional diffusion processes, (BiBoS preprint series). [15] Y. Okabe, On long time tails of correlation functions for KMO-Langevin equations, Probability Theory and Mathematical Statistics (S. Watanabe and Yu. V. Prokhorov, ed.), Lecture Notes in Math., 1299, Springer, 1988, pp. 391-397. [16] E. Seneta, Regular Varying Functions, Lecture Notes in Math., 508, Springer, 1976. [17] M. Tomisaki, Power order decay of elementary solutions of generalized diffusion equations, Probability Theory and Mathematical Statistics (S. Watanabe and Yu. V. Prokhorov, ed.), Lecture Notes in Math., 1299, Springer, 1988, pp. 511-523. [18] S. Watanabe, On time inversion of one-dimensional diffusion processes, Z. Wahrsch. verw. Geb., 31 (1975), 115-124. [19] K. Yoshida, Lectures on Differential and Integral Equations, Interscience, New York, 1960.
Date of correction: October 20, 2006Reason for correction: -Correction: PDF FILEDetails: -