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.
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References
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)
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. 39–69
J.-M. Bismut, Hypoelliptic Laplacian and Orbital Integrals (Princeton University Press, Princeton, 2011)
L. Hörmander, Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)
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. 281–305
A. Kolmogorov, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung). Ann. Math. 35(2), 116–117 (1934)
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. 195–263
H. Matsumoto, S. Taniguchi, Wiener functionals of second order and their Lévy measures. Electron. J. Probab. 7(14), 1–30 (2002)
J. Pitman, M. Yor, Infinitely divisible laws associated with hyperbolic functions. Can. J. Math. 55, 292–330 (2003)
M. Yor, Some Aspects of Brownian Motion. Part I: Some Special Functionals (Birkhäser, Boston, 1992)
M. Yor, Some Aspects of Brownian Motion. Part II: Some Recent Martingale Problems (Birkhäser, Boston, 1997)
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).
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Ikeda, N., Matsumoto, H. (2015). The Kolmogorov Operator and Classical Mechanics. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) In Memoriam Marc Yor - Séminaire de Probabilités XLVII. Lecture Notes in Mathematics(), vol 2137. Springer, Cham. https://doi.org/10.1007/978-3-319-18585-9_21
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DOI: https://doi.org/10.1007/978-3-319-18585-9_21
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