Time reversal of Markov processes and relativistic quantum theory

Chaos, Solitons and Fractals

Volumn 8, Issue 11, 1997, Pages 1711-1772

  Nagasawa, Masao ^a  

^a UNIVERSITY OF ZURICH   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

BROWNIAN MOVEMENT; CONVERGENCE OF NUMERICAL METHODS; DIFFUSION; ELECTROMAGNETIC FIELD EFFECTS; ELEMENTARY PARTICLES; MARKOV PROCESSES; MATHEMATICAL MODELS; NONLINEAR EQUATIONS; RELATIVITY;

RELATIVISTIC QUANTUM THEORY; SCHRODINGER EQUATION; SPINLESS PARTICLES;

QUANTUM THEORY;

EID: 0031270223     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0960-0779(97)00028-3     Document Type: Article

References (80)

1. Schrödinger, E., Über die Umkehrung der Naturgesetze. Sitzungs berichte der preussischen Akad. der Wissenschaften Physicalisch Mathematische Klasse, 1931, pp. 144-153.
2. Kolmogoroff, A., Zur Theorie der Markoffschen Ketten. Math. Ann., 1936, 112, 155-160.
3. Kolmogoroff, A., Zur Umkehrbarkeit der statistischen Naturgesetze. Math. Ann., 1937, 113, 766-772.
4. Hunt, G. A., Markov processes and potentials I, II, and III, Illinois J. Math., 1957, 1, 44-93, 316-369; 1958, 2, 151-213.
5. Hunt, G. A., Markov processes and potentials I, II, and III, Illinois J. Math., 1957, 1, 44-93, 316-369; 1958, 2, 151-213.
6. Hunt, G. A., Markov chains and Martin boundaries. Illinois J. Math., 1960, 4, 316-340.
7. Nagasawa, M., Time reversal of Markov processes. Nagoya Math. J., 1964, 24, 177-204.
8. Ikeda, N., Nagasawa, M. and Sato, K., A time reversal of Markov Processes with killing. Kodai Math. Sem. Rep, 1964, 16, 88-97.
9. Kunita, H. and Watanabe, T., On certain reversed processes and their applications to potential theory and boundary theory. J. Math. Mech., 1966, 15, 398-434.
10. Chung, K. L. and Walsh, J. B., To reverse a Markov process. Acta Math., 1969, 123, 225-251.
11. Meyer, P. A., Le retournement du temps, d'après Chung et Walsh. Springer Lecture Notes in Mathematics, Vol. 191, 1971, pp. 213-245.
12. Azema, J. Theorie générale des processus et retournement du temps. Ann. Sci. École Norm. Sup., 1973, 6, 459-519.
13. Nagasawa, M. and Maruyama, T., An application of time reversal of Markov processes to a problem of population genetics. Adv. in Appl. Probab., 1979, 11, 457-478.
14. Dynkin, E. B., An application of flows to time shift and time reversal in stochastic processes. Trans. Amer. Math. Soc., 1985, 287, 613-619.
15. Nagasawa, M., The adjoint process of a diffusion process with reflecting barrier. Kodai Math. Sem. Rep., 1961, 13, 235-248.
16. Föllmer, H., Time reversal on Wiener space. Bibos-Symposium 'Stochastic Processes in Math. Phys.', Springer Lecture Notes in Mathematics, Vol. 1158, 1986, 119-129.
17. Föllmer, H. and Wakolbinger, A., Time reversal of infinite dimensional diffusions. Stochastic Process. Appl., 1986, 22, 59-77.
18. Nagasawa, M., Schrödinger Equations and Diffusion Theory. Birkhäuser, Basel, Boston, Berlin, 1993.
19. Domenig, T. and Nagasawa, M., A Skorokhod problem with singular drift and its application to the origin of universes. Proc. Japan Acad. Ser. A, 1964, 70, 88-93.
20. Meyer, P. A., Sur une transformation du mouvement brownien due à Jeulin et Yor. Sém. de Probab. XXVIII, 1995.
21. Nagasawa, M. and Domenig, T., Diffusion processes on an open time interval and their time reversal. In Itô's Stochastic Calculus and Probability Theory, ed. Ikeda, Watanabe, Fukushima and Kunita. Springer, 1996, pp. 261-280.
22. Kuznezov, S. E., Construction of Markov processes with random birth and death times. Theor. Veryat. i Prim. (English translation Theory Probab. Appl.), 1973, 18, 596-601.
23. Getoor, R. K. and Glover, J., Riesz decomposition in Markov process theory. Trans. Amer. Math. Soc., 1984, 285, 107-132.
24. Getoor, R. K. and Sharpe, M. J., Naturality, standardness, and weak duality for Markov processes. Z. Wahrsch. Verw. Gebiete, 1984, 67, 1-62.
25. Sharpe, M. J., General Theory of Markov Processes. Academic Press, Boston, San Diego, New York, 1988.
26. Nagasawa, M., Segregation of a population in an environment. J. Math. Biol., 1980, 9, 213-235.
27. Nagasawa, M., Macroscopic, intermediate, microscopic and mesons. Lect. Notes Phys., 1985, 262, 427-437, Springer.
28. Nagasawa, M., Transformations of diffusion and Schrödinger processes. Probab. Theory Related Fields, 1989, 82, 109-136.
29. Nagasawa, M., Can the Schrödinger equation be a Boltzmann equation? Evanston, 1989. In Diffusion processes and Related problems in Analysis, ed. M. Pinsky. Birkhäuser, Basel, Boston, Berlin, 1990.
30. Nagasawa, M., The equivalence of diffusion and Schrödinger equations: A solution to Schrödinger's conjecture. Proceedings of Locarno Conference, 1991, ed. S. Albeverio. World Scientific, Singapore, 1991.
31. Nagasawa, M., A principle of superposition and interference of diffusion processes. Third European Symposium on Analysis and Probability, 1992, at the Institute Henri Poincaré, Sém. de Probabilités XXVII. Springer Lecture Notes in Mathematics, Vol. 1557, 1993, pp. 1-14.
32. Bochner, S., Diffusion equations and stochastic processes. Proc. Nat. Acad. Sci. USA, 1949, 35, 368-370.
33. Phillips, R. S., On the generation of semigroups of linear operators. Pacific J. Math., 1952, 2, 343-369.
34. Sato, K., Subordination depending on a parameter. In Probability Theory and Mathematical Statistics, Proc. Fifth Vilnius Conference, Vol. 2, 1990, 372-382.
35. Ichinose, T. and Tamura, H., Propagation of Dirac particle: A path integral approach. J. Math. Phys., 1984, 25, 1810-1819.
36. Ichinose, T., and Tamura, H., Imaginary-time path integral for relativistic spinless particle in an electromagnetic field. Comm. Math. Phys., 1986, 105, 239-257.
37. Ichinose, T., The non-relativistic limit problem for a relativistic spinless particle in an electromagnetic field. J. Func. Anal., 1987, 73, 233-257.
38. Ichinose, T., Remarks on the Weyl quantized relativistic Hamiltonian. Note di Mathematica XII, 1992, pp. 49-67.
39. Ichinose, T., Some results on the relativistic Hamiltonians: Selfadjointness and imaginary-time path integral. Proc. International Conf. Univ. Alabama, 1994.
40. Nagasawa, M., and Tanaka, H., A diffusion process in a singular meandriftfield. Z. Wahrsch. Verw. Gebiete, 1985, 68, 247-269.
41. Nagasawa, M., and Tanaka, H., Propagation of chaos for diffusing particles of two types with singular mean field interaction. Probab. Theory Related Fields, 1986, 71, 69-83.
42. Nagasawa, M., and Tanaka, H. Diffusion with interactions and collisions between coloured particles and the propagation of chaos. Probab. Theory Related Fields, 1987, 74, 161-198.
43. Nagasawa, M., and Tanaka, H. A proof of the propagation of chaos for diffusion processes with drift coefficients not of average form. Tokyo J. Math., 1987, 10, 403-418.
44. Nagasawa, M., Mathematical Foundations of Quantum Mechanics. Lecture Notes at the University of Zürich, 1994.
45. Nagasawa, M., Quantum theory, theory of Brownian motions, and relativity theory. Chaos, Solitons and Fractals, 1996, 7, 631-643.
46. Bochner, S., Diffusion equations and stochastic processes. Proc. Nat. Acad. Sci. USA, 1949, 35, 368-370.
47. Bochner, S., Harmonic Analysis and Theory of Probability. University of California Press, Berkeley, Los Angeles, 1960.
48. Sato, K., Stochastic Processes with Stationary Independent Increments (Japanese). Kinokuniya Shoten, Tokyo, 1990.
49. Sato, K., Lévy processes on the Euclidean spaces. Lecture Notes at the University of Zürich, 1995.
50. Bauer, H., Probability Theory and Elements of Measure Theory. Academic Press, London, New York, Toronto, Sidney, San Francisco, 1981.
51. Blumenthal, R. M. and Getoor, R. K., Markov Processes and Potential Theory. Academic Press, New York and London, 1968.
52. Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion. Springer, Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, Barcelona, 1991.
53. Jamison, B., Reciprocal processes. Z. Wahrsch. Verw. Gebiete, 1974, 30, 65-86.
54. Jamison, B., The Markov processes of Schrödinger. Z. Wahrsch. Verw. Gebiete, 32, 1975, 323-331.
55. Aebi, R., Schrödinger Diffusion Processes. Birkhäuser, Basel, Boston, Berlin, 1996.
56. Lamb, W. E., An operational interpretation of nonrelativistic quantum mechanics. Physics Today, 1969, 22(4), 23-28.
57. Nelson, E., Derivation of Schrödinger equation from Newtonian mechanics. Phys. Rev., 1966, 150, 1076-1085.
58. Aebi, R., Diffusions with singular drift related to wave functions. Probab. Theory Related Fields, 1993, 96, 107-121.
59. Bertoin, J., Lévy Processes. Cambridge University Press, 1996.
60. Yosida, K., On the differentiability and representation of one-parameter semi-group of linear operators. J. Math. Soc. Japan, 1948, 1, 15-21.
61. Itô, K., On stochastic processes (infinite divisible laws of probability) (Doctoral thesis). Japan. J. Math., XVIII, 1942, 261-301.
62. Dynkin, E. B., Markov Processes. Springer, Berlin, Göttingen, Heidelberg, 1965.
63. Vershik, A. and Yor, M., Multiplicativité du processus gamma et étude asymptotique des lois stables d'indice α, lorsque α tend vers 0. Prepublications du Lab. de probab. de l'universite Paris, VI, 1995, 284.
64. Nagasawa, M. and Tanaka, H., Stochastic differential equations of pure-jumps (to be published).
65. Kunita, H. and Watanabe, S., On square integrable martingales. Nagoya Math. J., 1967, 30, 209-245.
66. Itô, K., On Stochastic Differential Equations. Memoirs of the AMS, Vol. 4, American Mathematical Society, 1951.
67. Skorokhod, A. V., Studies in the Theory of Random Processes. Addison-Wesley, Reading, Mass, 1965.
68. McKean, H. P., A class of Markov processes associated with non-linear parabolic equations. Proc. Nat. Acad. Sci. USA, 1966, 56, 1907-1911.
69. Aebi, R. and Nagasawa, M., Large deviations and the propagation of chaos for Schrödinger processes. Probab. Theory Related Fields, 1992, 94, 53-68.
70. Dirac, P. A. M., The Principle of Quantum Mechanics. 1st and 4th ed., Oxford, New York, 1930 and 1958.
71. Einstein, A., Podolsky, B. and Rosen, N., Can quantum-mechanical description of physical reality be considered complete?. Phys. Rev., 1935, 47, 777-780.
72. Bell, J. S., On the Einstein Podolsky Rosen paradox. Physics, 1964, 1, 195-202.
73. Nagasawa, M., On the locality of hidden-variable theories in quantum physics. Chaos, Solitons and Fractals (in this volume).
74. Petrosky, T. and Prigogine, I., Poincaré resonances and extention of classical dynamics. Chaos, Solitons and Fractals, 1996, 7, 441-497.
75. El Naschie, M. S., Time symmetry breaking, duality and Cantorian space-time. Chaos, Solitons and Fractals, 1996, 7, 499-518.
76. Ord, G. N., Fractal space-time and statistical mechanics of random walks. Chaos, Solitons and Fractals, 1996, 7, 821-845.
77. Nottale, L., Scale relativity, fractal space-time and quantum mechanics. In Quantum Mechanics, Diffusion and Chaotic Fractals, ed. M. S. El Naschie, O. E. Rösler and I. Prigogine. Pergamon Press, Oxford, 1995, pp. 51-78.
78. Nottale, L., Scale relativity and fractal space-time: Application to quantum physics, cosmology and chaotic systems. Chaos, Solitons and Fractals, 1996, 7, 877-938.
79. Gudder, S., Hyperfinite quantum random walks. Chaos, Solitons and Fractals, 1996, 7, 669-679.
80. Garbaczewski, P., Klauder, J. R. and Olkiewicz, R., Schrödinger problem, Lévy processes, and noise in relativistic quantum mechanics. Phys. Rev. E, 1995, 51, 4114-4131.