no. 3
Probability and quantum symmetries. I. The theorem of Noether in Schrödinger's euclidean quantum mechanics
Annales de l'I.H.P. Physique théorique, Tome 67 (1997) no. 3, pp. 297-338. 
@article{AIHPA_1997__67_3_297_0,
     author = {Thieullen, M. and Zambrini, J. C.},
     title = {Probability and quantum symmetries. {I.} {The} theorem of {Noether} in {Schr\"odinger's} euclidean quantum mechanics},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     pages = {297--338},
     publisher = {Gauthier-Villars},
     volume = {67},
     number = {3},
     year = {1997},
     mrnumber = {1472821},
     zbl = {0897.60062},
     language = {en},
     url = {http://archive.numdam.org/item/AIHPA_1997__67_3_297_0/}
}
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Thieullen, M.; Zambrini, J. C. Probability and quantum symmetries. I. The theorem of Noether in Schrödinger's euclidean quantum mechanics. Annales de l'I.H.P. Physique théorique, Tome 67 (1997) no. 3, pp. 297-338. http://archive.numdam.org/item/AIHPA_1997__67_3_297_0/

[1] V. Arnold, Méthodes mathématiques de la mécanique classique, Ed. Mir, Moscou, 1976. | MR | Zbl

[2] V.M. Alekseev, V.M. Tikhomirov and S.V. Fomin, Commande Optimale, Ed. Mir, Moscou, 1982. | MR

[3] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and path integrals, Mc Graw - Hill, N. Y., 1965. | Zbl

[4] B. Simon, Functional Integration and Quantum Physics, Acad. Press, N. Y., 1979. | MR | Zbl

[5] N. Ikeda and S. Watanabe, Stochastic Differential equations and Diffusion Processes, 2nd ed., North Holland, Amsterdam, 1989. | MR | Zbl

[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. | MR | Zbl

[7] R.H. Cameron, J. Math. Physics, Vol. 39, 1960, p. 126. | Zbl

[8] E. Cartan, Leçons sur les invariants intégraux, Hermann, Paris, 1922. | MR

[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. | MR

[10] E. Nelson, Quantum fluctuations, Princeton Series in Physics, P. U. Press, 1985. | MR | Zbl

[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. | MR

[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. | MR | Zbl

[13] P. Ramond, Field Theory. A modern primer, Benjamin Cummings Publ., Reading, Mass., 1981. | MR

[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. | MR

[15] S. Weinberg, Annals of Physics, Vol. 194, 1989, p. 336. | MR

[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. | MR | Zbl

[17] A. Galindo and P. Pascual, Quantum Mechanics I, Springer-Verlag, 1989. | Zbl

[18] E. Nelson, Dynamical Theories of Brownian Motion, Princeton Univ. Press, N. J., 1967. | MR | Zbl

[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. | MR

[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. | MR | Zbl

[21] J. Doob, Bull. Soc. Math. de France, Vol. 85, 1957, p. 431. | Numdam | MR | Zbl

[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. | Zbl

[23] B. Djehiche and T. Kolsrud, Canonical transformations for diffusions, C.R. Acad. Sc. Paris, T. 321, Série I, 1995, pp. 339-344. | MR | Zbl

[24] P.L. Lions, Act. Math., Vol. 161, 1988, p. 243. | Zbl

[25] M. Thieullen and J.C. Zambrini, Symmetries in the Stochastic Calculus of Variations, to appear in Prob. Theory Related Fields. | MR | Zbl

[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. | MR

[27] M. Thieullen, Prob. Theory Related Fields, No. 97, 1993, p. 231. | MR

[28] P. Malliavin, Stochastic calculus of variations and hypo-elliptic operators, Proc. Int. Symp. SDE Kyoto, 1976, 1978, Kinokuniya, Tokyo. | Zbl

[29] E.B. Dynkin, Markov processes, Springer-Verlag, Berlin, 1965.

[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. | MR | Zbl

[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. | MR | Zbl

[32] J.C. Zambrini, Calculus of Variations and Quantum Probability in Lect. Notes in Control and Info., 1988, No. 121, p. 173, N. Y. | MR

[33] A.J. Krener, Reciprocal diffusions in flat space, Prob. Theory Related fields, No. 107, 1997, p. 243. | MR | Zbl

[34] B.C. Levy and A.J. Krener, Stochastic mechanics of reciprocal diffusions, J. Math. Physics, Vol. 37, 1996, p. 769. | MR | Zbl

[35] N.V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Reidel, Dodrecht, 1987.

[36] S. Albeverio, J. Rezende and J.C. Zambrini, Probability and quantum symmetries, II. The Theorem of Noether in quantum mechanics.

[37] T. Misawa, Conserved quantities and symmetry for stochastic dynamical systems., Phys. Letters A, 1994, No. 195, p. 185. | MR | Zbl

[38] P. Malliavin, Stochastic analysis, Grund. der Math. Wiss., Vol. 313, Springer, 1997. | Zbl