# American Institute of Mathematical Sciences

2013, 2013(special): 807-813. doi: 10.3934/proc.2013.2013.807

## Stochastic deformation of classical mechanics

 1 Grupo de Física Matemática, Instituto para a Investigação Interdisciplinar da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, PT-1649-003 Lisboa, Portugal

Received  September 2012 Published  November 2013

We describe a method of stochastic deformation of classical mechanics preserving the time symmetry of this theory. It provides a new general strategy to deform stochastically Geometric Mechanics.
Citation: Jean-Claude Zambrini. Stochastic deformation of classical mechanics. Conference Publications, 2013, 2013 (special) : 807-813. doi: 10.3934/proc.2013.2013.807
##### References:
 [1] J. M. Bismut, "Mécanique Aléatoire",, Springer-Verlag, (1981). Google Scholar [2] Xin Chen and A. B. Cruzeiro, Stochastic geodesics and stochastic backward equations on Lie groups,, in these proceedings., (). Google Scholar [3] K. L. Chung and J.-C. Zambrini, "Introduction to Random Time and Quantum Randomness",, World Scientific, (2003). Google Scholar [4] A. B. Cruzeiro and J.-C. Zambrini, Malliavin calculus and Euclidean quantum mechanics. I. Functional calculus,, J. Funct. Analysis, 96 (1991), 62. Google Scholar [5] R. P. Feynman and A. R. Hibbs, "Quantum Mechanics and Path Integrals",, McGraw-Hill, (1965). Google Scholar [6] W. H. Fleming and H. Mete Soner, "Controlled Markov Processes and Viscosity Solutions", $2^{nd}$ edition,, Springer, (2006). Google Scholar [7] N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes",, North Holland, (1981). Google Scholar [8] J. A. Lázaro-Camí and J. P. Ortega, Stochastic Hamiltonian dynamical systems,, Rep. Math. Phys., 61 (2008), 65. Google Scholar [9] C. Leonard, From the Schrödinger problem to the Monge-Kantorovich problem,, J. Funct. Anal., 262 (2012), 1879. Google Scholar [10] P. Lescot and J.-C. Zambrini, Probabilistic deformation of contact geometry, diffusion processes and their quadrature,, Progress in Probability, 59 (2007), 203. Google Scholar [11] N. Privault and J.-C. Zambrini, Markovian bridges and reversible diffusion processess with jumps,, Ann. Inst. H. Poincaré, 40 (2004), 599. Google Scholar [12] E. Schrödinger, Sur la theorie relativiste de l'électron et l'interprétation de la mécanique quantique,, Ann. Inst. H. Poincaré, 2 (1932), 269. Google Scholar [13] M. Thieullen and J.-C. Zambrini, Probability and quantum symmetries I. The theorem of Nœther in Schrödinger's Euclidean quantum mechanics},, Ann. Inst. H. Poincaré, 67 (1997), 297. Google Scholar [14] G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian Motion,, Physical Review, 36 (1930), 823. Google Scholar [15] P. A. Vuillermot and J.-C. Zambrini, Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations,, Journal of Theoretical Probability, (2012), 10959. Google Scholar [16] J.-C. Zambrini, Variational processes and stochastic versions of mechanics,, J. Math. Physics, 27(9) (1986), 2307. Google Scholar [17] J.-C. Zambrini, On the geometry of the Hamilton-Jacobi-Bellman equation,, Journal of Geometric Mechanics, 1 (2009), 369. Google Scholar [18] J.-C. Zambrini, The research program of stochastic deformation (with a view toward Geometric Mechanics),, , (). Google Scholar

show all references

##### References:
 [1] J. M. Bismut, "Mécanique Aléatoire",, Springer-Verlag, (1981). Google Scholar [2] Xin Chen and A. B. Cruzeiro, Stochastic geodesics and stochastic backward equations on Lie groups,, in these proceedings., (). Google Scholar [3] K. L. Chung and J.-C. Zambrini, "Introduction to Random Time and Quantum Randomness",, World Scientific, (2003). Google Scholar [4] A. B. Cruzeiro and J.-C. Zambrini, Malliavin calculus and Euclidean quantum mechanics. I. Functional calculus,, J. Funct. Analysis, 96 (1991), 62. Google Scholar [5] R. P. Feynman and A. R. Hibbs, "Quantum Mechanics and Path Integrals",, McGraw-Hill, (1965). Google Scholar [6] W. H. Fleming and H. Mete Soner, "Controlled Markov Processes and Viscosity Solutions", $2^{nd}$ edition,, Springer, (2006). Google Scholar [7] N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes",, North Holland, (1981). Google Scholar [8] J. A. Lázaro-Camí and J. P. Ortega, Stochastic Hamiltonian dynamical systems,, Rep. Math. Phys., 61 (2008), 65. Google Scholar [9] C. Leonard, From the Schrödinger problem to the Monge-Kantorovich problem,, J. Funct. Anal., 262 (2012), 1879. Google Scholar [10] P. Lescot and J.-C. Zambrini, Probabilistic deformation of contact geometry, diffusion processes and their quadrature,, Progress in Probability, 59 (2007), 203. Google Scholar [11] N. Privault and J.-C. Zambrini, Markovian bridges and reversible diffusion processess with jumps,, Ann. Inst. H. Poincaré, 40 (2004), 599. Google Scholar [12] E. Schrödinger, Sur la theorie relativiste de l'électron et l'interprétation de la mécanique quantique,, Ann. Inst. H. Poincaré, 2 (1932), 269. Google Scholar [13] M. Thieullen and J.-C. Zambrini, Probability and quantum symmetries I. The theorem of Nœther in Schrödinger's Euclidean quantum mechanics},, Ann. Inst. H. Poincaré, 67 (1997), 297. Google Scholar [14] G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian Motion,, Physical Review, 36 (1930), 823. Google Scholar [15] P. A. Vuillermot and J.-C. Zambrini, Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations,, Journal of Theoretical Probability, (2012), 10959. Google Scholar [16] J.-C. Zambrini, Variational processes and stochastic versions of mechanics,, J. Math. Physics, 27(9) (1986), 2307. Google Scholar [17] J.-C. Zambrini, On the geometry of the Hamilton-Jacobi-Bellman equation,, Journal of Geometric Mechanics, 1 (2009), 369. Google Scholar [18] J.-C. Zambrini, The research program of stochastic deformation (with a view toward Geometric Mechanics),, , (). Google Scholar
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