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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
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show all references

References:
[1]

Am. J. Phys., 36 (1968), p280. doi: 10.1119/1.1974504.  Google Scholar

[2]

second edition graduate texts in mathematics, 60, Springer-verlag, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[3]

Lecture notes in mathematics, 866, Springer, 1981.  Google Scholar

[4]

Compte rendu Mathématique, 342 (2006), 333-346. doi: 10.1016/j.crma.2005.12.028.  Google Scholar

[5]

Springer Proceedings in Mathematics and Statistics, 100 (2014), 163-184. doi: 10.1007/978-3-319-11292-3_6.  Google Scholar

[6]

Lect. Notes in Math., Springer, 1362 (1988), 101-123. doi: 10.1007/BFb0086180.  Google Scholar

[7]

North Holland, Amsterdam (Kodansha Ltd., Tokyo), 1981.  Google Scholar

[8]

Lect.Notes in Math., 463 Springer, 1975.  Google Scholar

[9]

Editions Mir Moscou U.R.S.S., 4th edition, 1988. Google Scholar

[10]

Rep. Math. Phys., 61 (2008), 65-112. doi: 10.1016/S0034-4877(08)80003-1.  Google Scholar

[11]

Discrete and Cont. Dyn. Systems A, 34 (2014), 1533-1574. doi: 10.3934/dcds.2014.34.1533.  Google Scholar

[12]

Probability Surveys, 11 (2014), 237-269. doi: 10.1214/13-PS220.  Google Scholar

[13]

Ann. Inst. H. Poincaré, 2 (1932), p269. Google Scholar

[14]

Ann. Inst. H.Poincaré, Phys. theo., 67 (1997), 297-338.  Google Scholar

[15]

Journal of Theoretical Probability, 27 (2014), 449-492. doi: 10.1007/s10959-012-0426-3.  Google Scholar

[16]

Physical Review A, 33 (1986), 1532-1548. doi: 10.1103/PhysRevA.33.1532.  Google Scholar

[17]

J. Math. Phys., 27 (1986), 2307-2330. doi: 10.1063/1.527002.  Google Scholar

[18]

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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