Abstract
We compute the small time asymptotics of the fundamental solution of Hörmander’s type hypoelliptic operators with drift, on the diagonal at a point x 0. We show that the order of the asymptotics depends on the controllability of an associated control problem and of its approximating system. If the control problem of the approximating system is controllable at x 0, then so is also the original control problem, and in this case we show that the fundamental solution blows up as \(\phantom {\dot {i}\!}t^{-\mathcal {N}/2}\), where \(\phantom {\dot {i}\!}\mathcal {N}\) is a number determined by the Lie algebra at x 0 of the fields, that define the hypoelliptic operator.
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Acknowledgments
The author is grateful with Andrei Agrachev for many useful discussions and for introducing to the studied problem, and with Davide Barilari, for his interest in the subject and for many illuminating questions and remarks. The author has been partially supported by the Institut Henri Poincaré, Paris, where part of this research has been carried out.
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Paoli, E. Small Time Asymptotics on the Diagonal for Hörmander’s Type Hypoelliptic Operators. J Dyn Control Syst 23, 111–143 (2017). https://doi.org/10.1007/s10883-016-9321-z
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DOI: https://doi.org/10.1007/s10883-016-9321-z