The Kolmogorov Operator and Classical Mechanics

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


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.


Zeta Function Heat Kernel Standard Brownian Motion Path Space Order Differential Operator 
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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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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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