American Institute of Mathematical Sciences

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

Stochastic deformation of classical mechanics

Jean-Claude Zambrini 1,

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.
Keywords: stochastic analysis., Stochastic control, geometric mechanics.
Mathematics Subject Classification: Primary: 60H10, 49L99, 70F9.
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, Berlin, 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), 62-95.  Google Scholar

[5]

R. P. Feynman and A. R. Hibbs, "Quantum Mechanics and Path Integrals", McGraw-Hill, New York, 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, Amsterdam, 1981.  Google Scholar

[8]

J. A. Lázaro-Camí and J. P. Ortega, Stochastic Hamiltonian dynamical systems, Rep. Math. Phys., 61 (2008), 65-122.  Google Scholar

[9]

C. Leonard, From the Schrödinger problem to the Monge-Kantorovich problem, J. Funct. Anal., 262 (2012), 1879-1920.  Google Scholar

[10]

P. Lescot and J.-C. Zambrini, Probabilistic deformation of contact geometry, diffusion processes and their quadrature, Progress in Probability, Birkhäuser, 59 (2007), 203-226.  Google Scholar

[11]

N. Privault and J.-C. Zambrini, Markovian bridges and reversible diffusion processess with jumps, Ann. Inst. H. Poincaré, (Probability and Statistics) 40 (2004), 599-633.  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), 269-310.  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, (Phys. Théor.) 67 (1997), 297-338.  Google Scholar

[14]

G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian Motion, Physical Review, 36 (1930), 823-841. 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, http://rd.springer.com/article/10.1007/s10959-012-0426-3, Springer-Verlag, 2012. Google Scholar

[16]

J.-C. Zambrini, Variational processes and stochastic versions of mechanics, J. Math. Physics, 27(9) (1986), 2307-2330.  Google Scholar

[17]

J.-C. Zambrini, On the geometry of the Hamilton-Jacobi-Bellman equation, Journal of Geometric Mechanics, 1 (2009), 369-387.  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, Berlin, 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), 62-95.  Google Scholar

[5]

R. P. Feynman and A. R. Hibbs, "Quantum Mechanics and Path Integrals", McGraw-Hill, New York, 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, Amsterdam, 1981.  Google Scholar

[8]

J. A. Lázaro-Camí and J. P. Ortega, Stochastic Hamiltonian dynamical systems, Rep. Math. Phys., 61 (2008), 65-122.  Google Scholar

[9]

C. Leonard, From the Schrödinger problem to the Monge-Kantorovich problem, J. Funct. Anal., 262 (2012), 1879-1920.  Google Scholar

[10]

P. Lescot and J.-C. Zambrini, Probabilistic deformation of contact geometry, diffusion processes and their quadrature, Progress in Probability, Birkhäuser, 59 (2007), 203-226.  Google Scholar

[11]

N. Privault and J.-C. Zambrini, Markovian bridges and reversible diffusion processess with jumps, Ann. Inst. H. Poincaré, (Probability and Statistics) 40 (2004), 599-633.  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), 269-310.  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, (Phys. Théor.) 67 (1997), 297-338.  Google Scholar

[14]

G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian Motion, Physical Review, 36 (1930), 823-841. 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, http://rd.springer.com/article/10.1007/s10959-012-0426-3, Springer-Verlag, 2012. Google Scholar

[16]

J.-C. Zambrini, Variational processes and stochastic versions of mechanics, J. Math. Physics, 27(9) (1986), 2307-2330.  Google Scholar

[17]

J.-C. Zambrini, On the geometry of the Hamilton-Jacobi-Bellman equation, Journal of Geometric Mechanics, 1 (2009), 369-387.  Google Scholar

[18]

J.-C. Zambrini, The research program of stochastic deformation (with a view toward Geometric Mechanics),, , ().   Google Scholar

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