Hypoelliptic Heat Kernel on Nilpotent Lie Groups

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Asaad, Malva
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The starting point of our analysis is an old idea of writing an eigenfunction expansion for a heat kernel considered in the case of a hypoelliptic heat kernel on a nilpotent Lie group. One of the ingredients we have is the generalized Fourier transform. The formula one gets using this approach is explicit as long as we can find all unitary irreducible representations of the group. In this thesis we consider a nilpotent Lie group of step n as an illustration of this technique. First we apply Kirillov’s orbit method to describe all unitary irreducible representations for the group. This allows us to write the corresponding hypoelliptic heat kernel using an integral formula over a Euclidean space. As an application, we describe a short-time behaviour of the hypoelliptic heat kernel in our case.