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).

[2]

Xin Chen and A. B. Cruzeiro, Stochastic geodesics and stochastic backward equations on Lie groups,, in these proceedings., ().

[3]

K. L. Chung and J.-C. Zambrini, "Introduction to Random Time and Quantum Randomness",, World Scientific, (2003).

[4]

A. B. Cruzeiro and J.-C. Zambrini, Malliavin calculus and Euclidean quantum mechanics. I. Functional calculus,, J. Funct. Analysis, 96 (1991), 62.

[5]

R. P. Feynman and A. R. Hibbs, "Quantum Mechanics and Path Integrals",, McGraw-Hill, (1965).

[6]

W. H. Fleming and H. Mete Soner, "Controlled Markov Processes and Viscosity Solutions", $2^{nd}$ edition,, Springer, (2006).

[7]

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

[8]

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

[9]

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

[10]

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

[11]

N. Privault and J.-C. Zambrini, Markovian bridges and reversible diffusion processess with jumps,, Ann. Inst. H. Poincaré, 40 (2004), 599.

[12]

E. Schrödinger, Sur la theorie relativiste de l'électron et l'interprétation de la mécanique quantique,, Ann. Inst. H. Poincaré, 2 (1932), 269.

[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.

[14]

G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian Motion,, Physical Review, 36 (1930), 823.

[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.

[16]

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

[17]

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

[18]

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

show all references

References:
[1]

J. M. Bismut, "Mécanique Aléatoire",, Springer-Verlag, (1981).

[2]

Xin Chen and A. B. Cruzeiro, Stochastic geodesics and stochastic backward equations on Lie groups,, in these proceedings., ().

[3]

K. L. Chung and J.-C. Zambrini, "Introduction to Random Time and Quantum Randomness",, World Scientific, (2003).

[4]

A. B. Cruzeiro and J.-C. Zambrini, Malliavin calculus and Euclidean quantum mechanics. I. Functional calculus,, J. Funct. Analysis, 96 (1991), 62.

[5]

R. P. Feynman and A. R. Hibbs, "Quantum Mechanics and Path Integrals",, McGraw-Hill, (1965).

[6]

W. H. Fleming and H. Mete Soner, "Controlled Markov Processes and Viscosity Solutions", $2^{nd}$ edition,, Springer, (2006).

[7]

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

[8]

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

[9]

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

[10]

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

[11]

N. Privault and J.-C. Zambrini, Markovian bridges and reversible diffusion processess with jumps,, Ann. Inst. H. Poincaré, 40 (2004), 599.

[12]

E. Schrödinger, Sur la theorie relativiste de l'électron et l'interprétation de la mécanique quantique,, Ann. Inst. H. Poincaré, 2 (1932), 269.

[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.

[14]

G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian Motion,, Physical Review, 36 (1930), 823.

[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.

[16]

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

[17]

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

[18]

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

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