Geodesic
Hypoelliptic heat kernel over $3$-step nilpotent Lie groups
Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Mathematical Control Theory and Mechanics (Suzdal, July 3–7, 2009), Tome 42 (2011), pp. 48-61

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In this paper, we provide explicitly the connection between the hypoelliptic heat kernel for some $3$-step sub-Riemannian manifolds and the quartic oscillator. We study the left-invariant sub-Riemannian structure on two nilpotent Lie groups, namely, the (2,3,4) group (called the Engel group) and the (2,3,5) group (called the Cartan group or the generalized Dido problem). Our main technique is noncommutative Fourier analysis, which permits us to transform the hypoelliptic heat equation into a one-dimensional heat equation with a quartic potential. 
@article{CMFD_2011_42_a5,
     author = {U. Boscain and J.-P. Gauthier and F. Rossi},
     title = {Hypoelliptic heat kernel over $3$-step nilpotent {Lie} groups},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {48--61},
     publisher = {mathdoc},
     volume = {42},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2011_42_a5/}
}
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U. Boscain; J.-P. Gauthier; F. Rossi. Hypoelliptic heat kernel over $3$-step nilpotent Lie groups. Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Mathematical Control Theory and Mechanics (Suzdal, July 3–7, 2009), Tome 42 (2011), pp. 48-61. http://geodesic.mathdoc.fr/item/CMFD_2011_42_a5/