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

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


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