SUB-LAPLACIANS WITH DRIFT
We define the (first order) correctors ip^n j q (cf. [BLP,
JKO]) as solutions of the problem
^ = - C i + Y
^') = 0.
If 1 j n then we just set ^ — 0.
Note that (Zibij) = 0, 1 i m and that by the property (6) in section 1.5.1,
(Cj) = 0, n j q. So the correctors are well defined.
Combining (1.5.3) and (1.5.4), we have that there is c 0 independent of /
(1.5-5) L(f+ £ *±.f\
= - Y (Cii-c*^'+ Y2
li,3Q V l£ra l£m /
with the function F satisfying \F(x,z)\
This expression motivates the following definitions:
The homogenised sub-Laplacian L# associated to L is defined to
be the operator
with coefficients defined by
Qtj = ( Cij ~
+ E ^ ^ ^ ) , 1 i,j q-
\ 0£m I
We define the (second order) correctors ^ u , 1 hj Q (cf.
[BLP, JKO]) as solutions of the problem
Combining the above defionitions with (1.5.5) we have the following expression
which explains the relation between the operators L and L#:
(1.5.6) l(f+ £ V ^ / + £ ^
dxi dxj J
with the function F satisfying \F(x, z)\ c
3. LH is an elliptic operator on
To prove this, it is enough to prove that
if^ = ( 6 , . . . , e n
) e R
, ? ^ 0 , t h e n