Probability and quantum symmetries. I. The theorem of Noether in Schrödinger's euclidean quantum mechanics

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1. [1] V. Arnold, Méthodes mathématiques de la mécanique classique, Ed. Mir, Moscou, 1976. Zbl0385.70001MR474391
2. [2] V.M. Alekseev, V.M. Tikhomirov and S.V. Fomin, Commande Optimale, Ed. Mir, Moscou, 1982. MR728225
3. [3] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and path integrals, Mc Graw - Hill, N. Y., 1965. Zbl0176.54902
4. [4] B. Simon, Functional Integration and Quantum Physics, Acad. Press, N. Y., 1979. Zbl0434.28013MR544188
5. [5] N. Ikeda and S. Watanabe, Stochastic Differential equations and Diffusion Processes, 2nd ed., North Holland, Amsterdam, 1989. Zbl0684.60040MR1011252
6. [6] M. Kac, On some connections between probability theory and differential and integral equations, Proc. of the 2nd, Berkeley Symp. on Prob. and Statistics, J. Newman Ed., Univ. of California Press, Berkeley, 1951. Zbl0045.07002MR45333
7. [7] R.H. Cameron, J. Math. Physics, Vol. 39, 1960, p. 126. Zbl0096.06901
8. [8] E. Cartan, Leçons sur les invariants intégraux, Hermann, Paris, 1922. MR355764
9. [9] a) J.C. Zambrini, J. Math. Physics, Vol. 27, 1986, p. 2307. b) S. Albeverio, K. Yasue and J.C. Zambrini, Ann. Inst. Henri Poincaré, Physique Théorique, Vol. 49, 1989, p. 259. MR854761
10. [10] E. Nelson, Quantum fluctuations, Princeton Series in Physics, P. U. Press, 1985. Zbl0563.60001MR783254
11. [11] J.C. Zambrini, An alternative starting point for Euclidean field theory: Euclidean Quantum Mechanics, IXth International Congress of Mathematical Physics, Swansea (U.K.), Adam Hilger, N. Y., 1989, p. 260. MR1033772
12. [12] A.B. Cruzeiro and J.C. Zambrini, Malliavin Calculus and Euclidean Quantum Mechanics. I. Functional Calculus, J. of Funct. Anal., Vol. 96, 1991, 1, p. 62. Zbl0733.35093MR1093507
13. [13] P. Ramond, Field Theory. A modern primer, Benjamin Cummings Publ., Reading, Mass., 1981. MR602697
14. [14] a) M.G. Crandall and P.L. Lions, Trans. Amer. M. S., No. 277, 1984, p. 1. b) W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, 1993. MR690039
15. [15] S. Weinberg, Annals of Physics, Vol. 194, 1989, p. 336. MR1015798
16. [16] T. Kolsrud and J.C. Zambrini, The mathematical framework of Euclidean Quantum Mechanics. An outline in "Stochastic Analysis and Applications", Ed. A. B. Cruzeiro and J. C. Zambrini, No. 26, 1991, Birkhäuser, Boston. Zbl0784.60106MR1168072
17. [17] A. Galindo and P. Pascual, Quantum Mechanics I, Springer-Verlag, 1989. Zbl0824.00008
18. [18] E. Nelson, Dynamical Theories of Brownian Motion, Princeton Univ. Press, N. J., 1967. Zbl0165.58502MR214150
19. [19] a) I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice-Hall, N.J., 1963. b) W. Dittrich and M. Reuter, Classical and Quantum Dynamics, Springer-Verlag, 1994. Zbl0127.05402MR160139
20. [20] a) W. MILLER Jr, Symmetry groups and their applications, Academic Press, N. Y., 1972. b) P.J. Olver, Applications of Lie groups to differential equations, Springer-Verlag, N. Y., 1986. Zbl0306.22001MR338286
21. [21] J. Doob, Bull. Soc. Math. de France, Vol. 85, 1957, p. 431. Zbl0097.34004MR109961
22. [22] a) B. Djehiche, J. of Math. Phys., Vol. 33, 1992, No. 9, p. 3050. b) B. Djehiche, Potential Analysis, Vol. 2, p. 349, 1993. Zbl0762.60100
23. [23] B. Djehiche and T. Kolsrud, Canonical transformations for diffusions, C.R. Acad. Sc. Paris, T. 321, Série I, 1995, pp. 339-344. Zbl0836.60084MR1346139
24. [24] P.L. Lions, Act. Math., Vol. 161, 1988, p. 243. Zbl0757.93082
25. [25] M. Thieullen and J.C. Zambrini, Symmetries in the Stochastic Calculus of Variations, to appear in Prob. Theory Related Fields. Zbl0868.60064MR1440139
26. [26] a) E. Schrödinger, Ann. Inst. H. Poincaré, Vol. 2, 1932, p. 269. b) S. Bernstein, Sur les liaisons entre les grandeurs aléatoires, Mathematikerkongr, Zurich, Band 1, 1932. c) B. Jamison and Z. Wahrscheinlich, Gebiete, Vol. 30, 1974, p. 65. Zbl0007.02104MR1508000
27. [27] M. Thieullen, Prob. Theory Related Fields, No. 97, 1993, p. 231. MR1240725
28. [28] P. Malliavin, Stochastic calculus of variations and hypo-elliptic operators, Proc. Int. Symp. SDE Kyoto, 1976, 1978, Kinokuniya, Tokyo. Zbl0411.60060
29. [29] E.B. Dynkin, Markov processes, Springer-Verlag, Berlin, 1965. Zbl0132.37901
30. [30] A.B. Cruzeiro and J.C. Zambrini, Malliavin Calculus and Euclidean Quantum Mechanics, II, J. of Funct. Analysis, Vol. 130, No. 2, 1995, p. 450. Zbl0821.60060MR1335388
31. [31] J.C. Zambrini, From Quantum Physics to Probability Theory and back, in "Chaos - the interplay between Stochastic and Deterministic behaviour", Springer lecture Notes in Physics, No. 457, GARBACZEWSKI et al., eds., Springer, Berlin, 1995. Zbl0839.60090MR1452626
32. [32] J.C. Zambrini, Calculus of Variations and Quantum Probability in Lect. Notes in Control and Info., 1988, No. 121, p. 173, N. Y. MR1231121
33. [33] A.J. Krener, Reciprocal diffusions in flat space, Prob. Theory Related fields, No. 107, 1997, p. 243. Zbl0868.60049MR1431221
34. [34] B.C. Levy and A.J. Krener, Stochastic mechanics of reciprocal diffusions, J. Math. Physics, Vol. 37, 1996, p. 769. Zbl0869.60090MR1371041
35. [35] N.V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Reidel, Dodrecht, 1987. Zbl0619.35004
36. [36] S. Albeverio, J. Rezende and J.C. Zambrini, Probability and quantum symmetries, II. The Theorem of Noether in quantum mechanics. Zbl1112.81072
37. [37] T. Misawa, Conserved quantities and symmetry for stochastic dynamical systems., Phys. Letters A, 1994, No. 195, p. 185. Zbl0941.34512MR1306235
38. [38] P. Malliavin, Stochastic analysis, Grund. der Math. Wiss., Vol. 313, Springer, 1997. Zbl0878.60001