Large time behavior for the heat equation on Carnot groups

TL;DR

This paper generalizes a function decomposition on Carnot groups and uses it to analyze the large-time behavior of solutions to the hypoelliptic heat equation, expressing solutions as sums of fundamental kernels and derivatives.

Contribution

It introduces a new decomposition of functions on Carnot groups and applies it to describe the asymptotic behavior of heat equation solutions.

Findings

Solutions decompose into fundamental kernels and derivatives at large times

The decomposition coefficients are moments of initial data

Provides a framework for understanding hypoelliptic heat flow on Carnot groups

Abstract

We first generalize a decomposition of functions on Carnot groups as linear combinations of the Dirac delta and some of its derivatives, where the weights are the moments of the function. We then use the decomposition to describe the large time behavior of solutions of the hypoelliptic heat equation on Carnot groups. The solution is decomposed as a weighted sum of the hypoelliptic fundamental kernel and its derivatives, the coefficients being the moments of the initial datum.