# American Institute of Mathematical Sciences

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

## A weak approach to the stochastic deformation of classical mechanics

 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.
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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