Hypoelliptic Heat Kernels on Nilpotent Lie Groups

Abstract

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 G. One of the ingredients of this approach 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 G. In the current paper we consider an n-step nilpotent Lie group Gn as an illustration of this techinique. First we apply Kirillov's orbit method to describe these representations for Gn. 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.

Explicit formulas for heat kernels on diamond fractals

“…A more general definition of a projective system of measure spaces can be found e.g. in [35,[2][3][4][5][6][7][8][9][10][11][12][13][14].…”

“…Even in more classical settings, the existence of the heat kernel is obtained in a fairly abstract way, where sometimes the kernel can be expressed in integral form; see e.g. [7,24,44]. Providing a formula like (1.1) or any other useful explicit expression is often a very difficult, if not an impossible task.…”

Uniform, localized asymptotics for sub-Riemannian heat kernels and diffusions

“…There are situations where, for more specialized sub-Riemannian structures, one can find expressions for the heat kernel that allow the small-time asymptotics to be extracted in an explicit way. For example, for left-invariant structures on Lie groups, generalized Fourier transforms can be used, as developed in [5]. The sub-Riemannian model spaces are especially well studied and have a large literature, but we mention [16] and [15] as two examples of explicit computation of the heat kernel and its small-time asymptotics on such spaces.…”