Abstract
We consider the heat kernel o t associated to the left invariant Laplacian with a drift term, on the affine group of the line. We obtain a large time upper estimate for o t .
Highlights
Let G be the two dimensional Lie group of affine transformations on IR, T : ~+ x (ç E l~) with 0 y = et E I~+ and t, x e R
This group is the only non abelian two dimensional Lie group, and it can be seen as the semidirect product ? X I~+ since
The Lie algebra g of G is spanned by the left-invariant vector fields and X = y~ ~x
Summary
The Lie algebra g of G is spanned by the left-invariant vector fields and X = y~ ~x. A left invariant distance d on G, called the control distance, is associated to these vector fields (cf [9]). Many authors have studied the heat kernel associated to a driftless Laplacian on various Lie groups and Riemannian manifolds (cf for instance [6], and references there, for an interesting survey). In the setting of Lie groups of polynomial growth, a large time upper estimate for the kernel has been obtained by Alexopoulos (cf [1]). In this paper we study the large time behavior of 03C6t on the affine group. (1.3) implies the following upper estimate for the kernel of the Laplacian (1.1): Theorem 1.2: There exists C;c > 0 such that.
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