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A Stochastic Look at Geodesics on the Sphere

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Geometric Science of Information (GSI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10589))

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Abstract

We describe a method allowing to deform stochastically the completely integrable (deterministic) system of geodesics on the sphere \(S^2\) in a way preserving all its symmetries.

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References

  1. Dubrovin, B.A., Krichever, J.M., Novikov, S.P.: Integrable systems I. In: Arnold, V.I., Novikov, S.P. (eds.) Dynamical Systems IV. Encyclopaedia of Mathematical Sciences, vol. 4, pp. 177–332. Springer, Heidelberg (1990)

    Chapter  Google Scholar 

  2. Olver, P.: Applications of Lie Groups to Differential Equations, 2nd edn. Springer, New York (1993)

    Book  MATH  Google Scholar 

  3. Itô, K.: The Brownian motion and tensor fields on a Riemannian manifold. In: Proceedings of the International Congress Mathematical (Stockholm), pp. 536–539 (1962)

    Google Scholar 

  4. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York (2005)

    MATH  Google Scholar 

  5. Malliavin, P.: Stochastic Analysis. Springer, Heidelberg (1997)

    Book  MATH  Google Scholar 

  6. Kuwabara, R.: On the symmetry algebra of the Schrödinger wave equation. Math. Japonica 22, 243 (1977)

    MATH  MathSciNet  Google Scholar 

  7. Léonard, C., Zambrini, J.-C.: Stochastic deformation of Jacobi’s integrability Theorem in Hamiltonian mechanics. Preparation

    Google Scholar 

  8. Zambrini, J.-C.: Probability and quantum symmetries in a riemannian manifold. Prog. Probab. 45 (1999). Birkhäuser

    Google Scholar 

  9. Kolsrud, T.: Quantum and classical conserved quantities: martingales, conservation law and constants of motion. In: Benth, F.E., Di Nunno, G., Lindstrøm, T., Øksendal, B., Zhang, T. (eds.) Stochastic Analysis and Applications: Abel Symposium, pp. 461–491. Springer, Heidelberg (2005)

    Google Scholar 

  10. Zambrini, J.-C.: The research program of Stochastic Deformation (with a view toward Geometric Mechanics). In: Dalang, R., Dozzi, M., Flandoli, F., Russo, F. (eds.) BirkhäuserStochastic Analysis, A Series of Lectures (2015)

    Google Scholar 

  11. Léonard, C.: A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. A 34(4) (2014)

    Google Scholar 

  12. Léonard, C., Roelly, S., Zambrini, J.-C.: Reciprocal processes. A measure-theoretical point of view. Probab. Surv. 11, 237–269 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  13. Arnold, V.: Méthodes Mathématiques de la Mécanique Classique. Mir, Moscou (1976)

    Google Scholar 

  14. Villani, C.: Optimal Transport, Old and New. Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg (2009)

    Book  MATH  Google Scholar 

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Correspondence to Marc Arnaudon .

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Arnaudon, M., Zambrini, JC. (2017). A Stochastic Look at Geodesics on the Sphere. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_55

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  • DOI: https://doi.org/10.1007/978-3-319-68445-1_55

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68444-4

  • Online ISBN: 978-3-319-68445-1

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