Estimate of an Hypoelliptic Heat-Kernel outside the Cut-Locus in Semi-Group Theory

Rémi Léandre

doi:10.4236/am.2012.312a285

Abstract

We give a proof in semi-group theory based on the Malliavin Calculus of Bismut type in semi-group theory and Wentzel-Freidlin estimates in semi-group of our result giving an expansion of an hypoelliptic heat-kernel outside the cut-locus.

Highlights

Let us consider some vector fields Xi,i 1, m on d with bounded derivatives at each order
We give a proof in semi-group theory based on the Malliavin Calculus of Bismut type in semi-group theory and Wentzel-Freidlin estimates in semi-group of our result giving an expansion of an hypoelliptic heat-kernel outside the cut-locus where Bismut’s non-degeneray condition plays a preominent role
The natural question is to know if the semi-group has an heat-kernel: Pt f x pt x, y f y dy (2) d

Summary

Introduction

Let us consider some vector fields Xi ,i 1, , m on d with bounded derivatives at each order. Let us suppose that the strong Hoermander hypothesis is checked: in such a case Hoermander ([1]) proved the existence of a smooth heat kernel. In particular [4] proved again the existence of the heat kernel by using the Malliavin Calculus of Bismut type in semi-group theory. For readers interested by short time asymptotics of heat-kernels by using probabilistic methods, we refer to the review papers [14,15,16] and to the book of Baudoin [17]. The object of this paper is to translate in semi-group theory the proof of Theorem 1 of Takanobu-Watanabe [11], by using the tools of stochastic analysis for estimate of heat kernels we have translated in semi-group theory in [21,22] and [23] for Varadhan type estimates

The Cut Locus Associated to a Sub-Riemannian Distance

Scheme of the Proof of Theorem 1

Proof of the Technical Lemmas