American Institute of Mathematical Sciences

Advanced
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).  doi: 10.1119/1.1974504.  Google Scholar

[2]

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

[3]

J.-M. Bismut, Mécanique Aléatoire,, Lecture notes in mathematics, (1981).   Google Scholar

[4]

J. Cresson and S. Darses, Plongement stochastique des systèmes Lagrangiens,, Compte rendu Mathématique, 342 (2006), 333.  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.  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., 1362 (1988), 101.  doi: 10.1007/BFb0086180.  Google Scholar

[7]

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

[8]

H. H. Kuo, Gaussian Measures in Banach Spaces,, Lect.Notes in Math., (1975).   Google Scholar

[9]

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

[10]

J. A. Lázaro-Cami and J. P. Ortega, Stochastic Hamiltonian dynamical systems,, Rep. Math. Phys., 61 (2008), 65.  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.  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.  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).   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, 67 (1997), 297.   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.  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.  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.  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, (2015).   Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

J.-M. Bismut, Mécanique Aléatoire,, Lecture notes in mathematics, (1981).   Google Scholar

[4]

J. Cresson and S. Darses, Plongement stochastique des systèmes Lagrangiens,, Compte rendu Mathématique, 342 (2006), 333.  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.  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., 1362 (1988), 101.  doi: 10.1007/BFb0086180.  Google Scholar

[7]

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

[8]

H. H. Kuo, Gaussian Measures in Banach Spaces,, Lect.Notes in Math., (1975).   Google Scholar

[9]

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

[10]

J. A. Lázaro-Cami and J. P. Ortega, Stochastic Hamiltonian dynamical systems,, Rep. Math. Phys., 61 (2008), 65.  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.  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.  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).   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, 67 (1997), 297.   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.  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.  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.  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, (2015).   Google Scholar

[1]

Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure & Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51
[2]

Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511
[3]

Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449
[4]

Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577
[5]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[6]

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295
[7]

TÔn Vı$\underset{.}{\overset{\hat{\ }}{\mathop{\text{E}}}}\, $T T$\mathop {\text{A}}\limits_. $, Linhthi hoai Nguyen, Atsushi Yagi. A sustainability condition for stochastic forest model. Communications on Pure & Applied Analysis, 2017, 16 (2) : 699-718. doi: 10.3934/cpaa.2017034
[8]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499
[9]

Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020002
[10]

Franco Flandoli, Dejun Luo. Euler-Lagrangian approach to 3D stochastic Euler equations. Journal of Geometric Mechanics, 2019, 11 (2) : 153-165. doi: 10.3934/jgm.2019008
[11]

Peter Brune, Björn Schmalfuss. Inertial manifolds for stochastic pde with dynamical boundary conditions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 831-846. doi: 10.3934/cpaa.2011.10.831
[12]

Ana Bela Cruzeiro. Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers. Journal of Geometric Mechanics, 2019, 11 (4) : 553-560. doi: 10.3934/jgm.2019027
[13]

Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343
[14]

Leonid Shaikhet. Improved condition for stabilization of controlled inverted pendulum under stochastic perturbations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1335-1343. doi: 10.3934/dcds.2009.24.1335
[15]

Marvin S. Müller. Approximation of the interface condition for stochastic Stefan-type problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4317-4339. doi: 10.3934/dcdsb.2019121
[16]

Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23
[17]

Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2081-2100. doi: 10.3934/cpaa.2020092
[18]

Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122
[19]

Jens Lang, Pascal Mindt. Entropy-preserving coupling conditions for one-dimensional Euler systems at junctions. Networks & Heterogeneous Media, 2018, 13 (1) : 177-190. doi: 10.3934/nhm.2018008
[20]

Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1441-1453. doi: 10.3934/dcdsb.2012.17.1441

2018 Impact Factor: 0.525

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences