A remark on the existence of a diffusion process with non-local boundary conditions

Yasushi ISHIKAWA

doi:10.2969/jmsj/04210171

Yasushi ISHIKAWA

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JOURNAL FREE ACCESS

1990 Volume 42 Issue 1 Pages 171-184

DOI https://doi.org/10.2969/jmsj/04210171

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Published: 1990 Received: January 09, 1989 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) R. F. Anderson, Diffusions with second order boundary conditions, Part I, II, Indiana Univ. Math. J., 25 (1976), 367-395, 403-411.
2) J.-M. Bony, P. Courrège et P. Priouret, Semi-group de Feller sur une varaété à bord compacte et problemes aux limites intégro-différentiels du second ordre donnant lieu au principle du maximum, Ann. Inst. Fourier (Grenoble), 18 (1968), 369-521.
3) L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math., 126 (1971), 11-51.
4) C. Cancelier, Problemes aux pseudo-differentiels donnant lieu au principle du maximium, Comm. P.D.E., 11 (1986), 1677-1726.
5) P. Cattiaux, Regularite au bord pour les densites et les densites conditionnelles d'une diffusion reflechie hypoelliptique, Stochastics, 20 (1987), 309-340,
6) P. Cattiaux, Stochastic calculus and degenerate boundary value problems, preprint, November 1987.
7) C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, In Conference on Harmonic Analysis (1981. Chicago), Wadsworth, Belmont, 1983, pp. 590-606.
8) L. Hörmander, The analysis of linear partial differential operators III, Springer, 1985.
9) N. Ikeda, On the construction of two-dimensional diffusion processes satisfying Wentzell's boundary conditions and its application to boundary value problems, Mem. Coll. Sci. Univ. Kyoto, Ser. A, 33 (1961), 367-427.
10) N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland/Kodansha, 1981.
11) T. Komatsu, On the martingale problem for generators of stable processes with perturbations, Osaka J. Math., 21 (1984), 113-132.
12) K. Sato and T. Ueno, Multi-dimensional diffusion and the Markov process on the boundary, J. Math. Kyoto Univ., 4 (1965), 529-605.
13) D. W. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. Pure Appl. Math., 24 (1972), 651-713.
14) K. Taira, Semigroups and boundary value problems, Duke Math. J., 49 (1982), 287-320.
15) K. Taira, Diffusion processes and partial differential equations, Academic Press, 1988.
16) S. Takanobu and S. Watanabe, On the existence and uniqueness of diffusion processes with Wentzell's boundary conditions, J. Math. Kyoto Univ., 28 (1988), 71-80.
17) S. Watanabe, On stochastic differential equations for multi-dimensional diffusion processes with boundary conditions I, II, J. Math. Kyoto Univ., 11 (1971), 169-180, 11(1971), 545-551.
18) S. Watanabe, Construction of diffusion processes with Wentzell's boundary conditions by means of Poisson point processes of Brownian excursions, Probability Theory, Banach Center Publications, 5, PWN-Polish Scientific Publishers, Warsaw, 1979, pp. 255-271.

Right : [1] R. F. Anderson, Diffusions with second order boundary conditions, Part I, II, Indiana Univ. Math. J., 25 (1976), 367-395, 403-411.
[2] J. -M. Bony, P. Courrège et P. Priouret, Semi-group de Feller sur une varaété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principle du maximum, Ann. Inst. Fourier (Grenoble), 18 (1968), 369-521.
[3] L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math., 126 (1971), 11-51.
[4] C. Cancelier, Problemes aux pseudo-differentiels donnant lieu au principle du maximium, Comm. P. D. E., 11 (1986), 1677-1726.
[5] P. Cattiaux, Regularite au bord pour les densites et les densites conditionnelles d'une diffusion reflechie hypoelliptique, Stochastics, 20 (1987), 309-340,
[6] P. Cattiaux, Stochastic calculus and degenerate boundary value problems, preprint, November 1987.
[7] C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, In Conference on Harmonic Analysis (1981. Chicago), Wadsworth, Belmont, 1983, pp. 590-606.
[8] L. Hörmander, The analysis of linear partial differential operators III, Springer, 1985.
[9] N. Ikeda, On the construction of two-dimensional diffusion processes satisfying Wentzell's boundary conditions and its application to boundary value problems, Mem. Coll. Sci. Univ. Kyoto, Ser. A, 33 (1961), 367-427.
[10] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland/Kodansha, 1981.
[11] T. Komatsu, On the martingale problem for generators of stable processes with perturbations, Osaka J. Math., 21 (1984), 113-132.
[12] K. Sato and T. Ueno, Multi-dimensional diffusion and the Markov process on the boundary, J. Math. Kyoto Univ., 4 (1965), 529-605.
[13] D. W. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. Pure Appl. Math., 24 (1972), 651-713.
[14] K. Taira, Semigroups and boundary value problems, Duke Math. J., 49 (1982), 287-320.
[15] K. Taira, Diffusion processes and partial differential equations, Academic Press, 1988.
[16] S. Takanobu and S. Watanabe, On the existence and uniqueness of diffusion processes with Wentzell's boundary conditions, J. Math. Kyoto Univ., 28 (1988), 71-80.
[17] S. Watanabe, On stochastic differential equations for multi-dimensional diflusion processes with boundary conditions I, II, J. Math. Kyoto Univ., 11 (1971), 169-180, 11 (1971), 545-551.
[18] S. Watanabe, Construction of diffusion processes with Wentzell's boundary conditions by means of Poisson point processes of Brownian excursions, Probability Theory, Banach Center Publications, 5, PWN-Polish Scientific Publishers, Warsaw, 1979, pp. 255-271.

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