Lie groups related to Hörmander operators and Kolmogorov-Fokker-Planck equations

Andrea Bonfiglioli; Ermanno Lanconelli

doi:10.3934/cpaa.2012.11.1587

Article Contents

2012, Volume 11, Issue 5: 1587-1614. Doi: 10.3934/cpaa.2012.11.1587

This issue Previous Article Next Article Positive solutions of a fourth-order boundary value problem involving derivatives of all orders

Lie groups related to Hörmander operators and Kolmogorov-Fokker-Planck equations

Andrea Bonfiglioli^1, and
Ermanno Lanconelli^1,

1.
Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato, 5 - 40126 Bologna

Received: January 2011

Revised: January 2012

Published: September 2012

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Abstract

If $\mathcal{L}=\sum_{j=1}^m X_j^2+X_0$ is a Hörmander partial differential operator in $\mathbb{R}^N$, we give sufficient conditions on the $X_j$'s for the existence of a Lie group structure $\mathbb{G}=(\mathbb{R}^N,*)$, not necessarily nilpotent, such that $\mathcal{L}$ is left invariant on $\mathbb{G}$. We also investigate the existence of a global fundamental solution $\Gamma$ for $\mathcal{L}$, providing results ensuring a suitable left invariance property of $\Gamma$. Examples are given for operators $\mathcal{L}$ to which our results apply: some are new, some appear in recent literature, usually quoted as Kolmogorov-Fokker-Planck type operators.

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Mathematics Subject Classification: Primary: 35J70, 43A80; Secondary: 37C10.

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