1. Introduction and statement of the results

Let G be a connected Lie group, let dg be a left invariant Haar measure on G

and let \A\ = dg-mea,sme(A), A being a Borel measurable subset of G.

Let us fix a compact neighborhood U of the identity element e of G and let

Un = {gig2-9n :gi,92,-,9n £ U}.

We set

(1.1)

\Q\G

= mf{n : g G Un}.

We shall assume that G has polynomial volume growth, i.e. there is a constant

c 0 such that

\Un\ cnc,n

G N. Then, (cf. [Gui]) there is an integer D 0,

such that

(1.2) -nD \Un\ cnD, n G N.

c

We call D homogeneous dimension (at infinity) of G. Note that D does not de-

pend on the choice of U. Note also that every connected nilpotent Lie group has

polynomial volume growth.

We identify the Lie algebra g of G with the left invariant vector fields on G.

By a left invariant sub-Laplacian on G, we shall understand an operator

L = -{El + ... + El) + E0

where

EQ,E\,...,EP

are left invariant vector fields and where the vector fields

E\,...,EP satisfy Hormander's condition, i.e. they generate together with their

successive Lie brackets [E^, [Ei2, [...,^fc]...]]], 1 i0 p, 1 j k the Lie algebra

flofG.

For results concerning the analysis of the symmetric sub-Laplacians (i.e. with

drift term £0 = 0) we reffer the reader to [VSC] (and the refferences therein).

In this paper, we are mainly interested in non-symmetric sub-Laplacians (i.e.

with EQ 7^ 0). We are interested, in particular, in the large time behavior of the heat

kernel pt(x,y) of L (i.e. the fundamental solution of the equation (J^ + L)u = 0).

Most of the techniques developped in [VSC] work only for symmetric operators.

So the presence of a drift term adds a new essential difficulty to the problem.

Our interest is also motivated by the central limit theorem. If we restrict our

interest to symmetric sub-Laplacians, then we can only study products of symmetric

random variables (with values on G). In order to deal with non-symmetric random

variables, we need to understand the large time behavior of the heat kernel of

non-symmetric sub-Laplacians.

Throughout this article, by the term simply connected we shall understand

connected and simply connected.

The different constants will always be denoted with the same letter c. When

their dependence or idependence is significant, it will be clearly stated.

Finally, we shall denote by ei the linear sub-space of $ generated by the vector

fields Ei,...,Ep and by t2 the linear sub-space of g generated by the Lie brackets

[EuEj], lijp.

Received by the editor February 10, 1997, and in revised form November 27, 1998.