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The Kolmogorov Operator and Classical Mechanics

  • Nobuyuki Ikeda
  • Hiroyuki Matsumoto
Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)

Abstract

The Kolmogorov operator is a quadratic differential operator which gives a typical example of a degenerate and hypoelliptic operator. The purpose of this note is to remark that the explicit expression for the transition probability density of the diffusion process generated by the Kolmogorov operator may be regarded as the Van Vleck formula. In fact, we show that it is given by the critical value of the action integral in some adequate path space.

Keywords

Zeta Function Heat Kernel Standard Brownian Motion Path Space Order Differential Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work is partially supported by Grants-in-Aid for Scientific Research (C) No.26400144 of Japan Society for the Promotion of Science (JSPS).

References

  1. 1.
    P. Biane, J. Pitman, M. Yor, Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull Am. Math. Soc. 38, 435–465 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    J.-M. Bismut, A survey of the hypoelliptic Laplacian, in Géométrie diffèrentielle, physique mathèmatique, mathèmatiques et société, II. Astérisque, vol. 322 (Société mathématique de France, Paris, 2008), pp. 39–69Google Scholar
  3. 3.
    J.-M. Bismut, Hypoelliptic Laplacian and Orbital Integrals (Princeton University Press, Princeton, 2011)zbMATHGoogle Scholar
  4. 4.
    L. Hörmander, Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    N. Ikeda, S. Kusuoka, S. Manabe, Lévy’s stochastic area formula and related problems, in Stochastic Analysis, ed. by M.Cranston, M. Pinsky. Proceedings of Symposia in Pure Mathematics, vol. 57 (American Mathematical Society, Providence, 1995), pp. 281–305Google Scholar
  6. 6.
    A. Kolmogorov, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung). Ann. Math. 35(2), 116–117 (1934)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, in Proceedings of the International Symposium on Stochastic Differential Equations, Kyoto 1976, ed. by K. Itô (Kinokuniya, Tokyo, 1978), pp. 195–263Google Scholar
  8. 8.
    H. Matsumoto, S. Taniguchi, Wiener functionals of second order and their Lévy measures. Electron. J. Probab. 7(14), 1–30 (2002)zbMATHMathSciNetGoogle Scholar
  9. 9.
    J. Pitman, M. Yor, Infinitely divisible laws associated with hyperbolic functions. Can. J. Math. 55, 292–330 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    M. Yor, Some Aspects of Brownian Motion. Part I: Some Special Functionals (Birkhäser, Boston, 1992)Google Scholar
  11. 11.
    M. Yor, Some Aspects of Brownian Motion. Part II: Some Recent Martingale Problems (Birkhäser, Boston, 1997)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Sanda, HyogoJapan
  2. 2.Department of Physics and MathematicsAoyama Gakuin UniversitySagamiharaJapan

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