for u, v ∈ Cyl, where Cyl is defined in 3.4 below. We shall prove that E , Cyl is closable and by classical Dirichlet form theory, we can obtain a probability measure-valued process. This process is
reversible w.r.t. the Ferguson-Dirichlet process Π
θ ,ν
. Moreover, we will show this process satisfies the Log-Sobolev inequality.
3 Construction of the Dirichlet form
3.1 Quasi-invariance property
In order to prove the symmetric bilinear form E defined in 2.14 is closable, we need to consider first the quasi-invariance of Π
θ ,ν
under the map e S
f
for f ∈ H . From now on for simplicity of
notation, we set P = P [0, 1] and Π
θ
= Π
θ ,Leb
that is, the Ferguson-Dirichlet measure Π
θ ,ν
with ν
= Lebesgue measure. It’s known that P is compact and complete under the weak topology, and its weak topology coincides with the topology determined by L
p
-Wasserstein distance d
w,p
, p ≥ 1, where
d
w,p
µ, ν = inf
π∈C µ,ν
Z
[0,1]
2
|x − y|
p
πdx, d y
1 p
. Here and in the sequel, C
µ, ν stands for the collection of all probability measures on [0, 1]×[0, 1] with marginals
µ and ν respectively. Set d
w
= d
w,2
. Refer to [23] for these fundamental results on probability measure space.
Recall that G denotes the space of all right continuous nondecreasing maps g : [0, 1] → [0, 1]. Each
g ∈ G can be extended to the full interval by setting g1 = 1. G
is equipped with L
2
-distance kg
1
− g
2
k
L
2
= Z
1
|g
1
t − g
2
t|
2
dt
1 2
.
Definition 3.1 For θ 0, there exists a unique probability measure Q
θ
on G , called Dirichlet process,
with the property that for each n ∈ N, and each family 0 = t t
1
. . . t
n
t
n+1
= 1, Q
θ
g
t
1
∈ dx
1
, . . . , g
t
n
∈ dx
n
= Γ
θ Π
n i=1
Γ θ t
i+1
− t
i n
Y
i=1
x
i+1
− x
i θ t
i+1
−t
i
−1
dx
1
· · · dx
n
, with x
n+1
= 1. The measure Q
θ
is sometimes called entropy measure, but as in [24], in this paper we use the entropy measure to only denote the push forward measure of Q
θ
under the map g 7→ g
∗
Le b. Define the map
ζ : G → P , g 7→ dg. It’s easy to see that
ζ
∗
Q
θ
:= Q
θ
◦ ζ
−1
= Π
θ
. Its inverse ζ
−1
assigns to each probability measure its distribution function. In the following we will study the quasi-invariance property of Π
θ ,ν
through Q
θ
and ζ. Von Renesse, M-K. and Sturm, K.T. [24] has
studied the quasi-invariance property of Q
θ
on G and under the map
χ : G → P , g 7→ g
∗
Le b, Q
θ
is pushed forward to a probability measure on P , which is called entropy measure there. Then through Dirichlet form theory, a stochastic process is constructed on P . Since its intrinsic metric of
this Dirichlet form is just the L
2
-Wasserstein distance on P , this process is usually called Wasserstein diffusion. Our present work also depends on the knowledge of Q
θ
. Let’s recall the quasi-invariance property of Q
θ
. 277
Theorem 3.2 [24] Theorem 4.3 Each C
2
-isomorphism h ∈ G induces a bijection map
τ
h
: G →
G , g 7→ h ◦ g, which leaves Q
θ
quasi-invariant: dQ
θ
h ◦ g = Y
θ h,0
gdQ
θ
g, and Y
θ h,0
is bounded from above and below. Here Y
θ h,0
g = X
θ h
gY
h,0
g, 3.1
where Y
h,0
g = 1
p h
′
0h
′
1 Y
a∈J
g
p h
′
ga
−
− h
′
ga
+ δh◦g
δg
a ,
X
h
g = exp θ
Z
1
log h
′
gsds ,
J
g
= x ∈ [0, 1]; gx
+
6= gx
−
, δh ◦ g
δg a =
hga
+
− hga
−
ga
+
− ga
−
. Due to the compactness of the interval [0, 1] and P , several well known topologies on G
and P coincide. More precisely, for each sequence g
n
⊂ G , and each g ∈ G
, the following types of convergence are equivalent:
• g
n
t → gt for each t ∈ [0, 1] in which g is continuous; • g
n
→ g in L
p
[0, 1] for each p ≥ 1; •
µ
g
n
→ µ
g
weakly; •
µ
g
n
→ µ
t
in the L
p
-Wasserstein distance for each p ≥ 1. Refer to [24] for a sketch of the idea of the argument.
Lemma 3.3 For each C
2
-isomorphism h ∈ G , ˜
τ
h
: P → P defined by 2.9 is continuous.
Proof. Let µ
n
∈ P , n ≥ 1 and µ
n
converges to µ as n → ∞. g
µ
n
and g
µ
denote the corre- sponding cumulative distribution functions of
µ
n
and µ. Then g
µ
n
converges to g
µ
in L
p
[0, 1] for each p ≥ 1.
Set ν
n
= ˜ τ
h
µ
n
, ν = ˜
τ
h
µ. Then the cumulative function of
ν
n
and ν
are h ◦ g
µ
n
and h ◦ g
µ
respectively. We denote by d
w,1
the L
1
-Wasserstein distance, that is, d
w,1
µ, ν = inf
π∈C µ,ν
n R
[0,1]
2
|x − y| πdx, d y
o . Then according to [22, Theorem 2.18] about
optimal transport on R, we have d
w,1
ν
n
, ν =
Z
1
|g
−1 ν
n
t − g
−1 ν
t|dt = Z
1
|g
ν
n
t − g
ν
t|dt =
Z
1
|h ◦ g
µ
n
t − h ◦ g
µ
t|dt ≤ max
s∈[0,1]
|h
′
s| Z
1
|g
µ
n
t − g
µ
t|dt, which yields that as
µ
n
weakly converges to µ, ν
n
converges to ν as well. This is the desired result.
In particular, ˜ τ
h
is measurable from P to P for C
2
-isomorphism h ∈ G .
278
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
