| Abstract: | Let  =(0, 1) be a fixed vector in R
2 with strictly positive components and suppose  0, 1 > 0. Set = 0 0 + 1 1 and, if x
0, x
1  R
n
, set x
= 0
x
0 + 1
x
1. Moreover, for any j {0, 1, }, let c
j
: R
n
 R be a continuous, bounded function and denote by p
j
, c
j
(t, x, y) the fundamental solution of the diffusion equation
If
then by applying the Girsanov transformation theorem of Wiener measure it is proved that
n
p
, c
(t, x
, y
)  {
n
0
p
0, c
0(t, x
0, y
0)}
0 0 / {
n
1
p
1, c
1(t, x
1, y
1)}
1 1 / for all x
0, x
0, y
0, y
1 R
n
and t > 0. Finally, in the last section, another proof of this inequality is given more in line with earlier investigations in this field. |
