Stochastic Differential Equations in a Differentiable Manifold

Kiyosi Itô

doi:10.1017/S0027763000022819

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The theory of stochastic differential equations in a differentiate manifold has been established by many authors from different view-points, especially by R Lévy [2], F. Perrin [1], A. Kolmogoroff [1] [2] and K. Yosida [1] [2]. It is the purpose of the present paper to discuss it by making use of stochastic integrals.

References

[1] Doob, J. L.: Stochastic processes depending on a continuous parameter, Trans. Amer. Math. Soc. 42, 1937.Google Scholar

[1] Ito, K.: Stochastic integral, Proc. Imp. Acad. Tokyo, Vol. 20, No. 8 (1944).Google Scholar

[2] Ito, K.: On a stochastic integral equation, Proc. Imp, Acad. Tokyo, Vol. 22, No. 2 (1946).Google Scholar

[3] Ito, K.: Stochastic differential equations, forthcoming in the Memoirs of Amer. Math. Soc. Google Scholar

[1] Kolmogoroff, A.: Zur Theorie der stetigen zufälligen Prozesse, Math. Ann. 108 (1933).CrossRef Google Scholar

[2] Kolmogoroff, A.: Umkehrbarkeit der stetigen Naturgestze, Math, Ann. 113 (1937).CrossRef Google Scholar

[1] Levy, P.: Theorie de l’addition des variables aléatoires, Paris (1937).Google Scholar

[2] Levy, P.: Processus stochastiques et mouvement brownien, Paris (1948).Google Scholar

[1] Perrin, F.: Étude mathematique du mouvement brownien de rotation, Ann. Éc. Norm., (3), 45 (1928).Google Scholar

[1] Yosida, K.: Brownian motion on the surface of the 3-sphere, Ann. of Math. Statistics, 20, 2 (1949).Google Scholar

[2] Yosida, K.: Integration of Fokker-Planck’s equation in a compact Riemannian space, Arkiv for Mathematik, 1, 9 (1949).Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Itô, Kiyosi 1950. Brownian motions in a Lie group. Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 26, Issue. 8,

Yosida, Kôsaku 1951. Integrability of the Backward Diffusion Equation in a Compact Riemannian Space. Nagoya Mathematical Journal, Vol. 3, Issue. , p. 1.

Hunt, G. A. 1956. Semi-groups of measures on Lie groups. Transactions of the American Mathematical Society, Vol. 81, Issue. 2, p. 264.

ITÔ, Seizô 1957. Fundamental solutions of parabolic differential equations and boundary value problems. Japanese journal of mathematics :transactions and abstracts, Vol. 27, Issue. 0, p. 55.

Petrov, V. V. 1959. Summary of Papers Presented at The Sessions of the Probability Research Seminar, Moscow University (February–May 1959). Theory of Probability & Its Applications, Vol. 4, Issue. 4, p. 432.

Khas’minskii, R. Z. 1960. Ergodic Properties of Recurrent Diffusion Processes and Stabilization of the Solution to the Cauchy Problem for Parabolic Equations. Theory of Probability & Its Applications, Vol. 5, Issue. 2, p. 179.

Motoo, Minoru 1960. Diffusion process corresponding to $$\frac{1}{2}\sum {\frac{{\partial ^2 }}{{\partial x^{i2} }}} + \sum {b^i (x)\frac{\partial }{{\partial x^i }}} $$. Annals of the Institute of Statistical Mathematics, Vol. 12, Issue. 1, p. 37.

Blagoveshchenskii, Yu. N. 1962. Diffusion Processes Depending on a Small Parameter. Theory of Probability & Its Applications, Vol. 7, Issue. 2, p. 130.

Freidlin, M. I. 1963. Diffusion Processes with Reflection and Problems with a Directional Derivative on a Manifold with a Boundary. Theory of Probability & Its Applications, Vol. 8, Issue. 1, p. 75.

McShane, E. J. 1963. Integrals devised for special purposes. Bulletin of the American Mathematical Society, Vol. 69, Issue. 5, p. 597.

TANAKA, Hiroshi 1964. EXISTENCE OF DIFFUSIONS WITH CONTINUOUS COEFFICIENTS. Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics, Vol. 18, Issue. 1, p. 89.

Gangolli, Ramesh 1964. On the construction of certain diffusions on a differentiable manifold. Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 2, Issue. 5, p. 406.

Freidlin, M. I. 1968. On the Factorization of Non-Negative Definite Matrices. Theory of Probability & Its Applications, Vol. 13, Issue. 2, p. 354.

Daletskii, Yu. L. and Shnaiderman, Ya. I. 1969. Diffusion and quasi-invariant measures on infinite-dimensional Lie groups. Functional Analysis and Its Applications, Vol. 3, Issue. 2, p. 156.

1969. Stochastic Integrals. p. 133.

Ku, Y.H. and Sheporaitis, L.P. 1970. Global properties of diffusion processes on cylindrical type phase space. Journal of the Franklin Institute, Vol. 289, Issue. 2, p. 93.

1972. Random Integral Equations. Vol. 96, Issue. , p. 218.

Kuo, Hui Hsiung 1972. Diffusion and Brownian motion on infinite-dimensional manifolds. Transactions of the American Mathematical Society, Vol. 169, Issue. 0, p. 439.

Sobko, G. M. 1973. The First Diffusion Problem on Differentiable Manifolds. Theory of Probability & Its Applications, Vol. 17, Issue. 3, p. 521.

1973. Summary of Papers Presented at Sessions of the Probability and Statistics Section of the Moscow Mathematical Society (September 1971–April 1972). Theory of Probability & Its Applications, Vol. 17, Issue. 4, p. 745.

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