Theorem 3.4 Quasi-invariance Let ν
be the Lebesgue measure on [0, 1]. For each C
2
-isomorphism h ∈ G
, the Ferguson-Dirichlet measure Π
θ ,ν
is quasi-invariant under the transformation ˜ τ
h
, and dΠ
θ ,ν
˜ τ
h
µ = Y
θ h,0
g
µ
dΠ
θ ,ν
µ, 3.2
where Y
θ h,0
g
µ
is defined as 3.1, and g
µ
denotes the cumulative distribution function of µ.
Proof. For any bounded measurable function u on P , it can induce a bounded measurable function
¯ u on G
by ¯
ug := u ζg.
Note that for C
2
-isomorphism h ∈ G ,
τ
−1 h
= τ
h
−1
see Theorem 3.2 for the definition, and ζ ◦ τ
h
◦ ζ
−1
= ˜ τ
h
. 3.3
So ˜ τ
h
is a bijection map and ˜ τ
−1 h
= ˜ τ
h
−1
. To see this, noting that g
µ
= ζ
−1
µ, we have for any f ∈ B [0, 1], for any
µ ∈ P , Z
1
f x d ζ ◦ τ
h
g
µ
= Z
1
f xd τ
h
g
µ
= Z
1
¯hg
µ
xdg
µ
x = Z
1
f x d ˜ τ
h
µ. According to Theorem 3.2, we obtain
Z
P
u µ dΠ
θ ,ν
˜ τ
h
µ = Z
P
u ˜ τ
−1 h
µ dΠ
θ ,ν
µ = Z
G
¯ u
τ
−1 h
g dQ
θ
g =
Z
G
¯ ugdQ
θ
τ
h
g = Z
G
¯ ugY
θ h,0
g dQ
θ
g =
Z
P
u µY
θ h,0
g
µ
dΠ
θ ,ν
µ, which concludes the proof.
3.2 Integration by parts formula
In section 2, we have defined the map e
t φ
: [0, 1] → [0, 1] and the derivative of a function u : P → R by
D
φ
u µ = lim
t→0
1 t
u ˜ τ
e
t φ
µ − uµ along
φ ∈ H , provided the limit exists. Let Cyl be the set of all functions on P in the form
u µ = F 〈 f
1
, µ〉, . . . , 〈 f
n
, µ〉,
3.4 where F ∈ C
1
R
n
, f
i
∈ C
1
[0, 1] for i = 1, . . . , n and n ∈ N. Let CylG be the set of all functions
w : G → R in the form
wg = F Z
f
1
dg, . . . , Z
f
n
dg 3.5
279
where F ∈ C
1
R
n
, f
i
∈ C
1
[0, 1] and n ∈ N. For a function w : G → R, define its directional
derivative along φ ∈ H
by D
φ
wg = d
dt
t=0
w τ
e
t φ
g = d
dt
t=0
we
t φ
◦ g 3.6
provided the limit exists.
Lemma 3.5 i For each w ∈ CylG in the form 3.5, D
φ
wg exists for each φ ∈ H
at every point g ∈ G
, and D
φ
wg =
n
X
i=1
∂
i
F Z
~fdg ·
Z
1
f
i
¯ φ
g
dg, 3.7
where R
~fdg = R
f
1
dg, . . . , R
f
n
dg and ¯ φ
g
x = R
1
φ
′
r gx
+
+ 1 − rgx
−
dr for g ∈ G .
ii For each u ∈ Cyl on P in the form 3.4, D
φ
u µ exists for each direction φ ∈ H
at every µ ∈ P ,
and D
φ
u µ =
n
X
i=1
∂
i
F 〈 ~ f ,
µ〉 · Z
1
f
i
x ¯ φ
g
µ
x d µx,
3.8 where 〈 ~
f , µ〉 = 〈 f
1
, µ〉, . . . , 〈 f
n
, µ〉 and ¯
φ
g
µ
defined as in i.
Proof. i By virtue of integration by parts formula on [0, 1], we have
Z
1
f
i
xdgx = f
i
1g1 − f
i
0g0 − Z
1
f
′ i
xgxdx. Using the chain rule for bounded variation function in Vol’pert average form cf. [1, Therem 3.96,
Remark 3.98], it holds d
dt
t=0
Z
1
f
i
xd e
t φ
◦ g x = −
Z
1
f
′ i
x φgx dx
= Z
1
f
i
x ¯ φ
g
x dgx by noting that
φ0 = φ1 = 0 for φ ∈ H . Therefore,
d dt
t=0
u τ
e
t φ
◦ g = d
dt
t=0
ue
t φ
◦ g = d
dt
t=0
F Z
~fxd e
t φ
◦ g x
=
n
X
i=1
∂
i
F Z
~fxdgx ·
Z
1
f
i
x ¯ φ
g
xdgx. ii Define ¯
ug = u ◦ ζg, then ¯
u is in the form ¯
ug = F Z
f
1
dg, . . . , Z
f
n
dg .
280
Invoking 3.3, d
dt
t=0
u ˜ τ
e
t φ
µ = d
dt
t=0
u ζτ
e
t φ
g
µ
= d
dt
t=0
¯ u
τ
e
t φ
g
µ
=
n
X
i=1
∂
i
F Z
~fdg
µ
· Z
1
f
i
x ¯ φ
g
µ
xdg
µ
x =
n
X
i=1
∂
i
F 〈 ~ f ,
µ〉 · Z
1
f
i
x ¯ φ
g
µ
xd µx.
which concludes the proof. Before stating the integration by parts formula, we recall a result on the derivative of g 7→ Y
θ h,0
g appeared in the quasi-invariance formula. According to [24, Lemma 5.7], for
φ ∈ C
2
[0, 1] with φ0 = φ1 = 0 and θ ≥ 0,
∂ ∂ t
Y
θ e
t φ
,0
g = X
a∈J
g
hφ
′
ga
+
+ φ
′
ga
−
2 −
φga
+
− φga
−
ga
+
− ga
−
i
+ θ
Z
1
φ
′
gx dx − φ
′
0 + φ
′
1 2
=: V
θ φ
g, 3.9
where J
g
= {x ∈ [0, 1]; gx
+
6= gx
−
}.
Theorem 3.6 Integration by parts formula i For each φ ∈ H
, u ∈ CylG , it holds that
Z
G
vgD
φ
ug dQ
θ
g = Z
G
ugD
∗ φ
vg dQ
θ
g 3.10
for any v ∈ CylG , where
D
∗ φ
vg = −D
φ
vg − V
θ φ
gvg. 3.11
ii For each φ ∈ H
, u ∈ Cyl, it holds that Z
P
v µD
φ
u µ dΠ
θ
µ = Z
P
u µD
∗ φ
v µ dΠ
θ
µ 3.12
for any v ∈ Cyl, where D
∗ φ
v µ = −D
φ
v µ − V
θ φ
g
µ
v µ.
3.13
Proof. We shall only prove i, and ii can be proved by the similar method used in the previous lemma. By the quasi-invariance of Q
θ
, one has Z
G
vgD
φ
ug dQ
θ
g = lim
t→0
1 t
Z
G
vgue
t φ
◦ g − ug dQ
θ
g = lim
t→0
1 t
Z
G
ug ve
−t φ
◦ gY
θ e
t φ
g − 1 dQ
θ
g =
Z
G
ug − D
φ
vg − V
θ φ
gvg dQ
θ
g,
281
which concludes i.
3.3 Tangent space and Dirichlet form
