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

Pierre Etoré (1) , Jose R León (2) , Clémentine Prieur (3)

1 IPS - Inférence Processus Stochastiques
2 UDELAR - Universidad de la República [Montevideo]
3 AIRSEA - Mathematics and computing applied to oceanic and atmospheric flows

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.

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Pierre Etoré, Jose R León, Clémentine Prieur. A probabilistic point of view for the exact or approximated computation of the solution to Kolmogorov hypoelliptic equations. 2021. ⟨hal-03342958v1⟩