Abstract
A general description of Bernstein processes, a class of diffusion processes, relevant to the probabilistic counterpart of quantum theory known as Euclidean Quantum Mechanics, is given. It is compatible with finite or infinite dimensional state spaces and singular interactions. Although the relations with statistical physics concepts (Gibbs measure, entropy,…) is stressed here, recent developments requiring Feynman’s quantum mechanical tools (action functional, path integrals, Noether’s Theorem,…) are also mentioned and suggest new research directions, especially in the geometrical structure of our approach.
Partially supported by ICCTI-CONICYT exchange program and the Presidential Chair on Qualitative Analysis of Quantum Dynamical Systems.
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References
J. C. Zambrini, Variational processes and stochastic versions of mechanics, J. Math. Phys. 27 (9) (1986), 2307.
S. Albeverio, K. Yasue, and J. C. Zambrini, Euclidean quantum mechanics: analytical approach, Ann. Inst. H. Poincaré (Phys. Théor.) 49 (3) (1989), 259.
E. Schrödinger, Sur la théorie relativiste de l’électron et l’interpretation de la mécanique quantique, Ann. Inst. H. Poincaré 2 (1932), 269.
H. Föllmer, Random fields and diffusion processes, in École d’Eté de St. Flour XV-XVII (1985–87), edited by P. L. Hennequin, Springer Lecture Notes in Mathematics 1362, Springer-Verlag, Berlin, 1988.
L. M. Wu, Feynman-Kac semigroups, ground state diffusions and large deviations, Journal of Funct. Anal. 123 (1994), p. 202.
D. Ruelle, Statistical Mechanics: Rigorous Results, Benjamin, New York, 1969.
B. Jamison, Reciprocal processes, Z. Wahrschernlichkeit V. Gebiete 30 (1974), p. 65.
A. B. Cruzeiro and J. C. Zambrini, J. of Funct. Anal. 96 (1) (1991), p. 62.
I. Csiszär, I-divergence geometry of probability distributions and minimization problems, Ann. Probability 3(1) (1975), 146.
M. Nagasawa, Schrödinger Equation and Diffusion Theory, Monographs in Math. 86 Birkhäuser, Basel, 1993.
M. Brunaud, Stoch. Proc. and Applic. 44 (1993), p. 329.
P. Cattiaux and Ch. Léonard, Minimization of Kullpack information for diffusion processes, Ann. Inst. H. Poincaré 30 (1) (1994), p. 83.
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965.
A. Beurling, An automorphism of product measures, Annals of Mathematics 72(1) (1960), p. 189.
C. Dellacherie and P. A. Meyer, Probabilités et Potentiel, vol. I, Hermann, Paris, 1975.
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag, New York, 1991.
E. Nelson, Quantum Fluctuations, Princeton Series in Physics (Princeton Univ. Press, 1985).
G. Jona-Lasinio and P. K. Mitter, Comm. Math. Physics 101 (1985), p. 409.
M. D. Donsker and S. R. S. Varadhan IV, Comm. Pure Appl. Math. 36 (1983), p. 183.
R. M. Blumenthal and R. K. Getoor, Markov Processes and their Potential Theory, Academic Press, New York, 1968.
W. A. Zheng, Ann. Inst. H. Poincaré 21 (1985), p. 103.
T. Kolsrud, Quantum constants of motion and the heat Lie algebra in a Riemannian manifold, Preprint, KTH, Stockholm,1996.
K. Itô, The Brownian motion and tensor fields on Riemannian manifold, in Proc. of Intern. Congress of Math., Stockholm, 1962, p. 536.
D. Dohrn and F. Guerra, Geodesic correction to stochastic parallel displacement of tensors, in Springer Lect. Notes in Phys. 93, edited by G. Casati and J. Ford, Springer-Verlag, Berlin, 1979, p. 241.
P. Malliavin, Stochastic Analysis, Grundleh. der Math. Wissen. 313, Springer-Verlag, Berlin, 1997.
W. H. Fleming and H. N. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993.
M. G. Crandall and P. Lions, Trans. Amer. Math. Soc. no. 277 (1984), p. 1.
J. C. Zambrini, Probability and quantum symmetries in a Riemannian manifold, in Progress in Probability Series, edited by R. Dalang, M. Dozzi and F. Russo, (1999).
P. J. Olver, Applications of Lie Groups to Differential Equations (New York: Springer-Verlag, 1986).
M. Thieullen and J. C. Zambrini, Ann. Inst. H. Poincaré (Phys. Théor.) 67 (3) (1997), p. 297.
M. Thieullen and J. C. Zambrini, Probability and quantum symmetries I. The theorem of Noether in Schrödinger’s Euclidean quantum mechanics, Prob. Theory and Related Fields 107 (1997), p. 401.
S. Albeverio, J. Rezende, and J. C. Zambrini, Probability and quantum symmetries II. The Theorem of Noether in quantum mechanics, in preparation.
W. Thirring, Quantum Mechanics of Atoms and Molecules, Springer-Verlag, Berlin, 1979.
J. L. McCauley, Classical Mechanics, Cambridge University Press, Cambridge, UK, 1997.
M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.
M. Sharpe, General Theory of Markov processes, Academic Press, New York, 1988.
M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Spaces and Symmetric Markov Processes, Walter de Gruyter, Berlin, 1994.
K. L. Chung and Z. Zhao, From Brownian Motion to Schrödinger’s Equation, Springer-Verlag, New York, 1994.
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Cruzeiro, A.B., Wu, L., Zambrini, J.C. (2000). Bernstein Processes Associated with a Markov Process. In: Rebolledo, R. (eds) Stochastic Analysis and Mathematical Physics. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1372-7_4
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DOI: https://doi.org/10.1007/978-1-4612-1372-7_4
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