The Kolmogorov Operator and Classical Mechanics


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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© 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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