SUB-LAPLACIANS WIT H DRIF T
3
1. Intuitive justification. When the drift term EQ G \) then its large time effect
is not important. For example, we can obtain a Gaussian estimate for the heat
kernel similar to the one known in the case when EQ = 0 (cf. [VSC]).
More precisely, let Ls = {E\ + ••• + E%) and let pf(x, y) be the heat kernel of
Ls. Then there is c 0 suet that (cf. [VI, V 3 , SC , VSC]) for all x, y G G, t 1
p f ( x , y ) c t - ^ 2 e x p ( - ^ - ^ )
pf(x,y) V D / 2 exp (
(i2-i)
-
c t e
^ i
*
We shall prove that the above estimates are still satisfied by the heat kernel
pt(x, y) of L = Ls + JE0, when £0 G f).
Intuitively, this can be explained as follows. Let us pretend, for simplification,
that EQ is also in the center of g. Then, we have that pt{x, e) = pf(exp(tEo)x, e).
So the existence of the drift term EQ implies that most of the mass of pt{x,e) lies
in a ball of radius y/i and centered at the point exp(—tEo) instead a ball of radius
\Jt and centered at the identity element e of G. If EQ G fy, then, grossly speaking,
we have to deal with some of following three cases:
1. EQ is in the Lie algebra of a compact Lie subgroup of G. Then there is c 0
such that
\ex.p(tE0)\G c , teR
and hence the estimates (1.2.1) are still valid.
2. G is nilpotent and EQ G [g,fl] (take for example the case of the Heisenberg
group). Then there is c 0 such that
| e x p ( ^
0
) | G c ^ , |*| 1 .
Hence, the estimates (1.2.1) are still valid.
3. This third case occurs when G is not nilpotent. The simplest example is the
the group of Euclidean motions on the plane. This group is isomorphic to
the semidirect product M2 X T, where T acts on R 2 by rotations. Note that
then H = R 2 X T. The drift E0 splits into two parts E0 = E0^R2 + E0j.
The part EQJ lies in T which is compact and hence by the case (1) has no
effect on the estimates (1.2.1). To treat the other term EQ^ which lies in
R 2 , we observe that L can be written as an operator on R 2 x T. Then,
Eo,R2 is a vector field with periodic coefficients in R 2 (cf. [Al , A2]). So,
what influences the large time behavior of the heat kernel, is not the drift
term £O,R2 itself, but the "average" or effective drift (cf. [JKO p. 68]). In
this particular example the effective drift vanishes. In the general case the
effective drift lies in the first commutator of some nilpotent Lie group and
hence, by (2), has no effect in the Gaussian estimates (1.2.1).
1.3. T h e passage from a non-centered t o a centered sub-Laplacian. If
L is not centered, then by conjugating L with a convenient multiplicative function
on G we can obtain a centered sub-Laplacian. Thus, a question about a non-
centered sub-Laplacian can be rephrased to a question about a centered one. More
presicely, let L = {E\ + ... + E"2) -f- EQ be a left invariant sub-Laplacian on a
connected Lie group G of polynomial volume growth and let us assume that L is
not centered, i.e. EQ £ I).
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