  In this work, we propose a method for solving Kolmogorov hypoelliptic
equations based on Fourier transform and Feynman-Kac formula. We first explain
how the Feynman-Kac formula can be used to compute the fundamental solution to
parabolic equations with linear or quadratic potential. Then applying these
results after a Fourier transform we deduce the computation of the solution to
a a first class of Kolmogorov hypoelliptic equations. Then we solve partial
differential equations obtained via Feynman-Kac formula from the
Ornstein-Uhlenbeck generator. Also, a new small time approximation of the
solution to Kolmogorov hypoelliptic equations is provided. We finally present
the results of numerical experiments to check the practical efficiency of this
approximation.
