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June  2016, 8(2): 221-233. doi: 10.3934/jgm.2016005

A weak approach to the stochastic deformation of classical mechanics

Rémi Lassalle 1, and  Jean Claude Zambrini 2,

1. 

Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France
2. 

GFM, Group of Mathematical Physics University of Lisbon, Department of Mathematics Faculty of Sciences, Campo Grande, Edifcio C6 PT-1749-016 Lisboa, Portugal

Received  December 2014 Revised  January 2016 Published  June 2016

We establish a transfer principle, providing a canonical form of dynamics to stochastic models, inherited from their classical counterparts. The stochastic deformation of Euler$-$Lagrange conditions, and the associated Hamiltonian formulations, are given as conditions on laws of processes. This framework is shown to encompass classical models, and the so-called Schrödinger bridges. Other applications and perspectives are provided.
Keywords: entropy., stochastic Hamilton conditions, Stochastic Euler-Lagrange condition.
Mathematics Subject Classification: Primary: 37N05, 60H30; Secondary: 60J6.
Citation: Rémi Lassalle, Jean Claude Zambrini. A weak approach to the stochastic deformation of classical mechanics. Journal of Geometric Mechanics, 2016, 8 (2) : 221-233. doi: 10.3934/jgm.2016005
References:
[1]

R. Abraham and J. E. Masden, Foundations of mechanics, Am. J. Phys., 36 (1968), p280. doi: 10.1119/1.1974504.  Google Scholar

[2]

V. I. Arnold, Mathematical methods of classical mechanics, second edition graduate texts in mathematics, 60, Springer-verlag, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[3]

J.-M. Bismut, Mécanique Aléatoire, Lecture notes in mathematics, 866, Springer, 1981.  Google Scholar

[4]

J. Cresson and S. Darses, Plongement stochastique des systèmes Lagrangiens, Compte rendu Mathématique, 342 (2006), 333-346. doi: 10.1016/j.crma.2005.12.028.  Google Scholar

[5]

A. B. Cruzeiro and R. Lassalle, On the least action principle for the Navier-Stokes equation, Springer Proceedings in Mathematics and Statistics, 100 (2014), 163-184. doi: 10.1007/978-3-319-11292-3_6.  Google Scholar

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H. Föllmer, Random fields and diffusion processes, École d' Été de Probabilités de Saint-Flour XV-XVII,1985-87 Lect. Notes in Math., Springer, 1362 (1988), 101-123. doi: 10.1007/BFb0086180.  Google Scholar

[7]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam (Kodansha Ltd., Tokyo), 1981.  Google Scholar

[8]

H. H. Kuo, Gaussian Measures in Banach Spaces, Lect.Notes in Math., 463 Springer, 1975.  Google Scholar

[9]

L. D. Landau and E. M. Lifshitz, Cours de Physique Théorique, Editions Mir Moscou U.R.S.S., 4th edition, 1988. Google Scholar

[10]

J. A. Lázaro-Cami and J. P. Ortega, Stochastic Hamiltonian dynamical systems, Rep. Math. Phys., 61 (2008), 65-112. doi: 10.1016/S0034-4877(08)80003-1.  Google Scholar

[11]

C. Leonard, A survey of the Schrödinger problem and some of its connections with optimal transport, Discrete and Cont. Dyn. Systems A, 34 (2014), 1533-1574. doi: 10.3934/dcds.2014.34.1533.  Google Scholar

[12]

C. Leonard, S. Roelly and J. C. Zambrini, Reciprocal processes: A measure-theoretical point of view, Probability Surveys, 11 (2014), 237-269. doi: 10.1214/13-PS220.  Google Scholar

[13]

E. Schrödinger, Sur la théorie relativiste de l'electron et l?interprétation de la mécanique quantique, Ann. Inst. H. Poincaré, 2 (1932), p269. Google Scholar

[14]

M. Thieullen and J. C. Zambrini, Probability and quantum symmetries I, the theorem of Noether in Schrödinger's euclidean quantum mechanics, Ann. Inst. H.Poincaré, Phys. theo., 67 (1997), 297-338.  Google Scholar

[15]

P. Vuillermot and J. C. Zambrini, Bernstein diffusions for a class of linear parabolic partial differential equations, Journal of Theoretical Probability, 27 (2014), 449-492. doi: 10.1007/s10959-012-0426-3.  Google Scholar

[16]

J. C. Zambrini, Stochastic mechanics according to E. Schrödinger, Physical Review A, 33 (1986), 1532-1548. doi: 10.1103/PhysRevA.33.1532.  Google Scholar

[17]

J. C. Zambrini, Variational processes and stochastic versions of mechanics, J. Math. Phys., 27 (1986), 2307-2330. doi: 10.1063/1.527002.  Google Scholar

[18]

J. C. Zambrini, The Research Program of Stochastic Deformation (with a View Toward Geometric Mechanics), Stochastic Analysis, a Series of lectures, Centre interfacultaire Bernouilli, EPFL, Program in Probability 68, Edit R.C. Dalang, M.Dozzi, F. Flandoli, F. Russo, Birkhäuser, 2015. Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Masden, Foundations of mechanics, Am. J. Phys., 36 (1968), p280. doi: 10.1119/1.1974504.  Google Scholar

[2]

V. I. Arnold, Mathematical methods of classical mechanics, second edition graduate texts in mathematics, 60, Springer-verlag, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[3]

J.-M. Bismut, Mécanique Aléatoire, Lecture notes in mathematics, 866, Springer, 1981.  Google Scholar

[4]

J. Cresson and S. Darses, Plongement stochastique des systèmes Lagrangiens, Compte rendu Mathématique, 342 (2006), 333-346. doi: 10.1016/j.crma.2005.12.028.  Google Scholar

[5]

A. B. Cruzeiro and R. Lassalle, On the least action principle for the Navier-Stokes equation, Springer Proceedings in Mathematics and Statistics, 100 (2014), 163-184. doi: 10.1007/978-3-319-11292-3_6.  Google Scholar

[6]

H. Föllmer, Random fields and diffusion processes, École d' Été de Probabilités de Saint-Flour XV-XVII,1985-87 Lect. Notes in Math., Springer, 1362 (1988), 101-123. doi: 10.1007/BFb0086180.  Google Scholar

[7]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam (Kodansha Ltd., Tokyo), 1981.  Google Scholar

[8]

H. H. Kuo, Gaussian Measures in Banach Spaces, Lect.Notes in Math., 463 Springer, 1975.  Google Scholar

[9]

L. D. Landau and E. M. Lifshitz, Cours de Physique Théorique, Editions Mir Moscou U.R.S.S., 4th edition, 1988. Google Scholar

[10]

J. A. Lázaro-Cami and J. P. Ortega, Stochastic Hamiltonian dynamical systems, Rep. Math. Phys., 61 (2008), 65-112. doi: 10.1016/S0034-4877(08)80003-1.  Google Scholar

[11]

C. Leonard, A survey of the Schrödinger problem and some of its connections with optimal transport, Discrete and Cont. Dyn. Systems A, 34 (2014), 1533-1574. doi: 10.3934/dcds.2014.34.1533.  Google Scholar

[12]

C. Leonard, S. Roelly and J. C. Zambrini, Reciprocal processes: A measure-theoretical point of view, Probability Surveys, 11 (2014), 237-269. doi: 10.1214/13-PS220.  Google Scholar

[13]

E. Schrödinger, Sur la théorie relativiste de l'electron et l?interprétation de la mécanique quantique, Ann. Inst. H. Poincaré, 2 (1932), p269. Google Scholar

[14]

M. Thieullen and J. C. Zambrini, Probability and quantum symmetries I, the theorem of Noether in Schrödinger's euclidean quantum mechanics, Ann. Inst. H.Poincaré, Phys. theo., 67 (1997), 297-338.  Google Scholar

[15]

P. Vuillermot and J. C. Zambrini, Bernstein diffusions for a class of linear parabolic partial differential equations, Journal of Theoretical Probability, 27 (2014), 449-492. doi: 10.1007/s10959-012-0426-3.  Google Scholar

[16]

J. C. Zambrini, Stochastic mechanics according to E. Schrödinger, Physical Review A, 33 (1986), 1532-1548. doi: 10.1103/PhysRevA.33.1532.  Google Scholar

[17]

J. C. Zambrini, Variational processes and stochastic versions of mechanics, J. Math. Phys., 27 (1986), 2307-2330. doi: 10.1063/1.527002.  Google Scholar

[18]

J. C. Zambrini, The Research Program of Stochastic Deformation (with a View Toward Geometric Mechanics), Stochastic Analysis, a Series of lectures, Centre interfacultaire Bernouilli, EPFL, Program in Probability 68, Edit R.C. Dalang, M.Dozzi, F. Flandoli, F. Russo, Birkhäuser, 2015. Google Scholar

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