A probabilistic point of view for the Kolmogorov hypoelliptic equations

TL;DR

This paper introduces a Fourier transform and Feynman-Kac based method for solving Kolmogorov hypoelliptic equations, providing new approximations and numerical validation for their solutions.

Contribution

It presents a novel approach combining Fourier transform and Feynman-Kac formula to solve a class of Kolmogorov hypoelliptic equations, including new small time approximations.

Findings

Effective numerical approximations demonstrated

New small time solution approximation proposed

Method validated through numerical experiments

Abstract

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.