A weak approach to the stochastic deformation of classical mechanics

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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

Abstract
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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.
[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.
[3]	J.-M. Bismut, Mécanique Aléatoire, Lecture notes in mathematics, 866, Springer, 1981.
[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.
[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.
[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.
[7]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam (Kodansha Ltd., Tokyo), 1981.
[8]	H. H. Kuo, Gaussian Measures in Banach Spaces, Lect.Notes in Math., 463 Springer, 1975.
[9]	L. D. Landau and E. M. Lifshitz, Cours de Physique Théorique, Editions Mir Moscou U.R.S.S., 4th edition, 1988.
[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.
[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.
[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.
[13]	E. Schrödinger, Sur la théorie relativiste de l'electron et l?interprétation de la mécanique quantique, Ann. Inst. H. Poincaré, 2 (1932), p269.
[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.
[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.
[16]	J. C. Zambrini, Stochastic mechanics according to E. Schrödinger, Physical Review A, 33 (1986), 1532-1548. doi: 10.1103/PhysRevA.33.1532.
[17]	J. C. Zambrini, Variational processes and stochastic versions of mechanics, J. Math. Phys., 27 (1986), 2307-2330. doi: 10.1063/1.527002.
[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.

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References:

[1]	R. Abraham and J. E. Masden, Foundations of mechanics, Am. J. Phys., 36 (1968), p280. doi: 10.1119/1.1974504.
[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.
[3]	J.-M. Bismut, Mécanique Aléatoire, Lecture notes in mathematics, 866, Springer, 1981.
[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.
[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.
[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.
[7]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam (Kodansha Ltd., Tokyo), 1981.
[8]	H. H. Kuo, Gaussian Measures in Banach Spaces, Lect.Notes in Math., 463 Springer, 1975.
[9]	L. D. Landau and E. M. Lifshitz, Cours de Physique Théorique, Editions Mir Moscou U.R.S.S., 4th edition, 1988.
[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.
[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.
[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.
[13]	E. Schrödinger, Sur la théorie relativiste de l'electron et l?interprétation de la mécanique quantique, Ann. Inst. H. Poincaré, 2 (1932), p269.
[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.
[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.
[16]	J. C. Zambrini, Stochastic mechanics according to E. Schrödinger, Physical Review A, 33 (1986), 1532-1548. doi: 10.1103/PhysRevA.33.1532.
[17]	J. C. Zambrini, Variational processes and stochastic versions of mechanics, J. Math. Phys., 27 (1986), 2307-2330. doi: 10.1063/1.527002.
[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.

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