# 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] 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

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
 [1] Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2725-3737. doi: 10.3934/dcds.2020383 [2] Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021009 [3] Sel Ly, Nicolas Privault. Stochastic ordering by g-expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 61-98. doi: 10.3934/puqr.2021004 [4] J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 [5] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [6] Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208 [7] María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 [8] Zhang Chen, Xiliang Li, Bixiang Wang. Invariant measures of stochastic delay lattice systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3235-3269. doi: 10.3934/dcdsb.2020226 [9] Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021079 [10] Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023 [11] Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201 [12] Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 [13] Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066 [14] Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209 [15] Yingxu Tian, Junyi Guo, Zhongyang Sun. Optimal mean-variance reinsurance in a financial market with stochastic rate of return. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1887-1912. doi: 10.3934/jimo.2020051 [16] Mrinal K. Ghosh, Somnath Pradhan. A nonzero-sum risk-sensitive stochastic differential game in the orthant. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021025 [17] Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020 [18] Jianxun Liu, Shengjie Li, Yingrang Xu. Quantitative stability of the ERM formulation for a class of stochastic linear variational inequalities. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021083 [19] Zheng Liu, Tianxiao Wang. A class of stochastic Fredholm-algebraic equations and applications in finance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3879-3903. doi: 10.3934/dcdsb.2020267 [20] Bing Gao, Rui Gao. On fair entropy of the tent family. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3797-3816. doi: 10.3934/dcds.2021017

2019 Impact Factor: 0.649

## Metrics

• PDF downloads (120)
• HTML views (0)
• Cited by (3)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]