Hypoelliptic heat kernel on nilpotent Lie groups

TL;DR

This paper develops an explicit integral formula for the hypoelliptic heat kernel on nilpotent Lie groups using the generalized Fourier transform and Kirillov's orbit method, and analyzes its short-time behavior.

Contribution

It introduces a method to explicitly compute the hypoelliptic heat kernel on n-step nilpotent Lie groups via representation theory and orbit method, providing new analytical tools.

Findings

Explicit integral formula for heat kernel on nilpotent Lie groups

Application of Kirillov's orbit method to describe representations

Analysis of short-time behavior of the heat kernel

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 $G_{n}$ as an illustration of this technique. First we apply Kirillov's orbit method to describe these representations for $G_{n}$. 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 behavior of the hypoelliptic heat kernel in our case.