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A probabilistic point of view for the exact or approximated computation of the solution to Kolmogorov hypoelliptic equations

Pierre Etoré 1 Jose León 2 Clémentine Prieur 3
1 IPS - Inférence Processus Stochastiques
LJK - Laboratoire Jean Kuntzmann
3 AIRSEA - Mathematics and computing applied to oceanic and atmospheric flows
Inria Grenoble - Rhône-Alpes, UGA - Université Grenoble Alpes, LJK - Laboratoire Jean Kuntzmann, Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology
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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03342958
Contributor : Pierre Etore Connect in order to contact the contributor
Submitted on : Monday, September 13, 2021 - 4:52:59 PM
Last modification on : Tuesday, October 19, 2021 - 11:25:56 AM

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  • HAL Id : hal-03342958, version 1
  • ARXIV : 2109.06529

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Pierre Etoré, Jose León, Clémentine Prieur. A probabilistic point of view for the exact or approximated computation of the solution to Kolmogorov hypoelliptic equations. 2021. ⟨hal-03342958⟩

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