| Issue |
ESAIM: PS
Volume 27, 2023
|
|
|---|---|---|
| Page(s) | 668 - 693 | |
| DOI | https://doi.org/10.1051/ps/2023007 | |
| Published online | 14 July 2023 | |
A probabilistic point of view for the Kolmogorov hypoelliptic equations
1
Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, LJK, Grenoble, France
2
Univ. de La República, IMERL, Montevideo, Uruguay, Escuela de Matemática UCV, Caracas, Venezuela
3
Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, LJK, Grenoble, France
* Corresponding author: rlramos@fing.edu.uy
Received:
17
August
2021
Accepted:
13
March
2023
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 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 a certain class of Kolmogorov hypoelliptic equations is provided. We finally present the results of numerical experiments to check the practical efficiency of this approximation.
Mathematics Subject Classification: 60H30 / 60H10 / 60J35
Key words: Hypoelliptic Kolmogorov equation / Feynman–Kac formula / asymptotic expansion
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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