Download Measurable metrics, intrinsic metrics and Lipschitz functions

Measurable metrics, intrinsic metrics and Lipschitz functions
Francis HIRSCH
Equipe d’Analyse et Probabilités
Université d’Évry - Val d’Essonne
Boulevard F. Mitterrand
F-91025 EVRY CEDEX
[email protected]
In the paper [29], N. Weaver introduced the notion of measurable metric and that of
Lipschitz function with respect to such a metric. The study of these notions was pursued by
the same author in subsequent papers ([30, 32, · · ·]) and in the book [31].
In our paper [16], we treated various important examples and, in particular, we studied
the intrinsic measurable metric associated with a local Dirichlet form and notably the case of
Wiener spaces.
On the other hand, M. Hino and J.A. Ramirez ([15]) showed that the intrinsic measurable
metric associated with a symmetric diffusion semigroup was strongly involved in the description of the Gaussian behavior in small time of such a semigroup. Moreover, we introduced in
[17] the concentration function related to a measurable metric and studied the corresponding
Gaussian concentration property.
In this paper, we shall give a survey of this set of results.
We first fix the framework and some notation.
All functions that we consider in this paper are supposed to be R-valued .
In what follows, we consider a σ-finite measure space (X, µ). Spaces L∞ or L2 are related
to this measure space.
If S is a set of classes (with respect to the µ-a.e. equality) of measurable
W functions, S has
a least upper bound or essential supremum which
is
a
class
denoted
by
S, and a greatest
V
lower bound or essential infimum denoted by S.
We denote by Ω the collection of all positive measure subsets of X. If A and B belong to Ω,
we denote by A ∼ B the fact that A and B only differ from a null set (µ(A\B) = µ(B\A) = 0).
If h is any function on X 2 and if A and B belong to Ω, we denote by
ess inf{h(x, y) : x ∈ A, x ∈ B}
the greatest C ∈ R such that C ≤ h(x, y) for µ-almost every x ∈ A and for µ-almost every
y ∈ B.
1
1
Measurable metrics
In [29], N. Weaver introduced the following definition.
Definition 1.1 A measurable pseudometric is a map ρ : Ω2 −→ [0, ∞] such that
1. A0 ∼ A =⇒ ρ(A0 , B) = ρ(A, B)
2. ρ(A, A) = 0
3. ρ(A, B) = ρ(B, A)
4. ρ(∪∞
n=1 An , B) = inf n≥1 ρ(An , B)
5. for any A, B, C,
ρ(A, C) ≤ supB 0 ⊂B (ρ(A, B 0 ) + ρ(B 0 , C))
where all subsets which appear above belong to Ω.
We now give two classes of examples which will be of special interest.
1.1
Example
Let ρ0 be a map from X 2 into [0, ∞] satisfying, for any x, y, z ∈ X,
ρ0 (x, x) = 0,
ρ0 (x, y) = ρ0 (y, x)
and ρ0 (x, z) ≤ ρ0 (x, y) + ρ0 (y, z).
Such a map ρ0 will be called a pointwise semimetric.
For A, B ∈ Ω, we set
ρ0 (A, B) = inf{ρ0 (x, y) : x ∈ A, y ∈ B},
and
ρ(A, B) = sup{ρ0 (A0 , B 0 ) : A0 ∼ A, B 0 ∼ B}.
We also have
ρ(A, B) = ess inf{ρ0 (x, y) : x ∈ A, y ∈ B}.
For A ⊂ X, we denote by ρA
0 the function
ρA
0 (x) = inf{ρ0 (x, y) : y ∈ A}.
Generally, ρ satisfies conditions 1 to 4 of Definition 1.1, but, without additional assumption,
condition 5 is not satisfied.
Proposition 1.2 ([16, Proposition 2.3.]) Suppose that ρ0 satisfies the following additional
assumption:
0
For all A ∈ Ω, there exists A0 ∈ Ω with A0 ⊂ A, A0 ∼ A and ρA
0 measurable.
Then ρ is a measurable pseudometric which will be called associated with ρ0 .
2
The hypothesis in the previous statement is in particular satisfied under the following assumptions: X is a topological space, ρ0 is a lower semicontinuous function on X × X, and µ is a
Borel measure which is inner regular, which means that, for any Borel set A,
µ(A) = sup{µ(K) : K compact and K ⊂ A}.
In particular, if (X, µ) satisfies these conditions and if Φ denotes a set of real continuous
functions on X, we can take
ρΦ
0 (x, y) = sup{|u(x) − u(y)| : u ∈ Φ}
and the associated measurable pseudometric will be denoted by ρΦ .
1.2
Example
We first introduce a notation which will be used in all what follows.
Let u be a µ-class of measurable real functions. If A ∈ Ω, we denote by FA (u) the support
of the image measure (uA )∗ (µA ), where the index A indicates the restriction to A. This
support FA (u) also is the essential image of the restriction uA of u to A. We set, for A, B ∈ Ω,
ρu (A, B) = d(FA (u), FB (u))
where we denote by d the usual distance in R:
ρu (A, B) = inf{|x − y| : x ∈ FA (u), y ∈ FB (u)}.
If we set, for a particular representative ũ of u,
ρũ0 (x, y) = |ũ(x) − ũ(y)|,
then ρu is nothing but the map ρ associated with ρũ0 by the method of the previous example
1.1:
ρu (A, B) = ess inf{|ũ(x) − ũ(y)| : x ∈ A, y ∈ B}.
It is easy to see that the map ρu is a measurable pseudometric.
Consider now a collection Ψ of classes and set
ρΨ (A, B) = sup{ρu (A, B) : u ∈ Ψ}.
Then, the map ρΨ satisfies properties 1, 2, 3 and 5 of Definition 1.1. But generally, condition
4 is not satisfied.
The following proposition is essentially [31, Example 6.2.5] (see [16]).
Proposition 1.3 Let F be a subspace of L∞ containing the constant function 1, and let k k0
be a seminorm on F such that k1k0 = 0. We assume that, for any M, N ≥ 0 , for any subset
S of the set
FM,N = {f ∈ F : kf kL∞ ≤ M and kf k0 ≤ N },
W
the essential supremum S belongs to FM,N . Then, denoting by Ψ the set {f ∈ F : kf k0 ≤ 1},
ρΨ is a measurable pseudometric.
3
In the proof of this result, the following important lemma, which is a reformulation of [31,
Lemma 6.2.4]) is useful.
Lemma 1.4 Let F be as above. Then, for any f ∈ F , for any A, B ∈ Ω, there exists g ∈ F
such that
kgk0 ≤ kf k0 , 0 ≤ g ≤ ρf (A, B), ρg (A, B) = ρf (A, B), g = 0 on A, g = ρf (A, B) on B.
2
Lipschitz functions
We henceforth consider a measurable pseudometric ρ, on the σ-finite measure space (X, µ).
Definition 2.1 Let u be a µ-class of measurable real functions. Then u is said to be a
ρ-Lipschitz function if there exists a constant C ≥ 0 such that
∀A, B ∈ Ω
ρu (A, B) ≤ C ρ(A, B).
In this case,
Lρ (u) = sup{ρu (A, B)/ρ(A, B) : A, B ∈ Ω and ρ(A, B) > 0}
is finite and called the Lipschitz constant of u.
We shall denote by Lip(ρ) the set of all ρ-Lipschitz functions, and by Lip∞ (ρ) (resp. Lip2 (ρ))
the set Lip(ρ) ∩ L∞ (resp. Lip(ρ) ∩ L2 ).
Definition 2.2 The measurable pseudometric ρ is called a measurable metric if Lip∞ (ρ) is
weak*-dense in L∞ .
For an equivalent definition, we refer to [32, Proposition 14].
The following statement comes from [31]. Notice that the proof is much less obvious than
that of the similar results for classical Lipschitz functions. Actually, the proof in [31] has to
be modified ([33]).
Proposition 2.3 ([31, Theorem 6.2.7]) Let F = Lip∞ (ρ) and, for f ∈ F , set kf k0 =
Lρ (f ). Then
1. F is a subalgebra of L∞ containing the constant function 1, and k
on F such that k1k0 = 0. Moreover, for any f, g ∈ F
kf gk0 ≤ kf kL∞ kgk0 + kgkL∞ kf k0 .
2. F equipped with the norm
kf kρ = kf kL∞ + kf k0
is a Banach space.
3. For any M, N ≥ 0 , for any subset S of the set
FM,N = {f ∈ F : kf kL∞ ≤ M and kf k0 ≤ N },
W
the essential supremum
S belongs to FM,N .
4
k0 is a seminorm
Consequently, the space F = Lip∞ (ρ) equipped with the seminorm k k0 = Lρ satisfies the
hypotheses of Proposition 1.3. Actually, we also have the following converse.
Proposition 2.4 ([16, Proposition2.10.]) Suppose that a subspace F of L∞ equipped with
a seminorm k k0 satisfies the hypotheses of Proposition 1.3. Then
F = Lip∞ (ρΨ )
and
∀f ∈ F
kfk0 = LρΨ (f),
where the measurable pseudometric ρΨ is that which is defined in the statement of Proposition
1.3.
In what follows, if B ⊂ X, we denote by 1B the indicator function of B.
The following result also comes from [31].
Proposition 2.5 ([31, Lemma 6.2.8]) Let A ∈ Ω and a > 0. Set
_
ρaA = {min(ρ(A, B), a) 1B : B ∈ Ω}.
Then ρaA ∈ Lip∞ (ρ) and Lρ (ρaA ) ≤ 1. Moreover, ρaA (x) = 0 almost everywhere on A.
We then get easily the following Corollary ([17]).
Corollary 2.6 For any A, B ∈ Ω,
ρ(A, B) = sup{ρf (A, B) : f ∈ Lip∞ (ρ) and Lρ (f ) ≤ 1}.
By previous Proposition 2.3 and Corollary 2.6, we therefore see that any measurable pseudometric may be defined by the method of Proposition 1.3.
We now introduce as in [17] the distance function to a subset.
Definition 2.7 Set, for A ∈ Ω,
ρA =
_
{ρ(A, B) 1B : B ∈ Ω}.
The class ρA is called the distance function to A.
It is easy to verify that, for any a > 0, ρaA = ρA ∧ a.
The characterization below shows in particular that this distance function coincides with
that introduced in [15] in the framework of local Dirichlet forms.
Proposition 2.8 ([17, Theorem 2.7]) Let A ∈ Ω. Then
_
ρA = {f : f ∈ Lip∞ (ρ), Lρ (f ) ≤ 1, f ≥ 0, f (x) = 0 a.e. on A}.
Moreover, ρA is the unique class ϕ satisfying the following properties:
1. ϕ ≥ 0 and ϕ(x) = 0 a.e. on A
2. ∀B ∈ Ω, ess inf{ϕ(x) : x ∈ B} = ρ(A, B)
5
3. ∀a > 0, (ϕ ∧ a) ∈ Lip(ρ) and Lρ (ϕ ∧ a) ≤ 1.
The following Proposition gives a necessary and sufficient condition in order to ρA itself be a
Lipschitz function.
Proposition 2.9 ([17, Proposition 2.8]) Let A ∈ Ω. Then ρA (x) < +∞ a.e. if and only
if, for any B ∈ Ω, ρ(A, B) < +∞. If this is satisfied, ρA ∈ Lip(ρ) and Lρ (ρA ) ≤ 1.
In fact, any ρ-Lipschitz function may be described in terms of the distance function as it is
shown in the following proposition (see [31, Theorem 6.2.9]).
Proposition 2.10 Let f ∈ Lip(ρ) and C ≥ Lρ (f ). Then
_
f = {f A − C ρA : A ∈ Ω},
where f A denotes the essential infimum of f (x) for x ∈ A.
We now consider the situation of Example 1.1. We assume that ρ0 is a pointwise semimetric satisfying the hypothesis of Proposition 1.2 and that ρ is the associated measurable
pseudometric. We then have:
Proposition 2.11 ([17, Proposition 2.9]) For A ∈ Ω,
_
0
0
0
A0
ρA = {ρA
0 : A ⊂ A, A ∼ A, ρ0 measurable}.
As a consequence, we have the following corollary.
0
Corollary 2.12 For any A ∈ Ω, there exists A0 ⊂ A such that A0 ∼ A, ρA
0 is measurable and
0
ρA
is
a
representative
of
ρ
.
A
0
We now recall an elementary definition.
Definition 2.13 A real valued function h is said to be ρ0 -Lipschitz continuous if h is measurable and if there exists C ≥ 0 such that
∀x, y ∈ X
|h(x) − h(y)| ≤ C ρ0 (x, y).
For such a function h, the ρ0 -Lipschitz constant Lρ0 (h) is the smallest constant C satisfying
the above inequalities.
A
Obviously, if for some A ∈ Ω ρA
0 is measurable, then for any a ≥ 0, ρ0 ∧ a is ρ0 -Lipschitz
continuous and its ρ0 -Lipschitz constant is ≤ 1. Therefore, we can obtain the following
consequence of Proposition 2.10 and Corollary 2.12.
Theorem 2.14 If f ∈ Lip∞ (ρ), there exists a bounded ρ0 -Lipschitz continuous representative
f˜ of f such that Lρ (f ) = Lρ0 (f˜). As a consequence, if f ∈ L∞ , f ∈ Lip∞ (ρ) if and only if f
has a ρ0 -Lipschitz continuous representative, and then
Lρ (f ) = min{Lρ0 (h) : h ρ0 -Lipschitz continuous representative of f }.
6
3
Dirichlet forms
We consider in this section a Dirichlet structure (X, B, µ, D, E) in the sense of [3] to which we
refer for the main definitions, properties and examples. In particular, µ is as before a σ-finite
measure on X equipped with the σ-algebra B, and E is a Dirichlet form on L2 whose domain
is D. We assume that the Dirichlet form E is local (which means e.g.
∀f, g ∈ D ∀a ∈ R
(f + a) g = 0 =⇒ E(f, g) = 0)
and there exists a carré du champ operator Γ (which means that there exists a continuous
map Γ : D × D −→ L1 such that
Z
∞
∀f, g, h ∈ D ∩ L
E(f h, g) + E(gh, f ) − E(f g, h) = h Γ(f, g) dµ).
In what follows, we write Γ(f ) instead of Γ(f, f ). We set
D∞ = {f ∈ D ∩ L∞ : Γ(f ) ∈ L∞ }.
All the results of this section come from [16].
3.1
Case 1 ∈ D
We first assume that the constant function 1 belongs to D. This implies of course that the
measure µ is finite.
Proposition 3.1 The space F = D∞ , equipped with the seminorm kf k0 = kΓ(f )1/2 kL∞
satisfies the assumptions of Proposition 1.3.
We now define the intrinsic measurable pseudometric ρ by
∀A, B ∈ Ω
ρ(A, B) = sup{ρf (A, B) : f ∈ D∞ and Γ(f ) ≤ 1}.
Then, by Proposition 1.3 and Proposition 2.4, we have:
Corollary 3.2 ρ is a measurable pseudometric, Lip∞ (ρ) = D∞ and, for any f ∈ D∞ ,
Lρ (f ) = kΓ(f )1/2 kL∞ .
We also have:
Corollary 3.3
Lip2 (ρ) = {f ∈ D : Γ(f ) ∈ L∞ }
and, for any f ∈ Lip2 (ρ), Lρ (f ) = kΓ(f )1/2 kL∞ . Moreover, for any A, B ∈ Ω,
ρ(A, B) = sup{ρf (A, B) : f ∈ D and Γ(f ) ≤ 1}.
7
3.2
General case
We no longer assume 1 ∈ D. However, we may use the following fact: There exists φ ∈ D
such that φ(x) > 0 µ-a.e. This is easy to see, using the σ-finiteness of µ and the density of D
in L2 . Let now ϕp = (p φ) ∧ 1 and Ap = {ϕp = 1}. Then ϕp ∈ D, 0 ≤ ϕp ≤ 1 and ∪p Ap = X
a.e. We set
e = {f ∈ L∞ : ∀ϕ ∈ D ϕ f ∈ D}.
D
e ⊂ Dloc , where the space Dloc of functions locally in D is defined in [3]: Actually, if
Then, D
e
f ∈ D, then f = ϕp f on Ap , ϕp f ∈ D and ∪p Ap = X a.e. By the hypothesis of locality, the
e ×D
e by setting
carré du champ operator can be extended to Dloc × Dloc and therefore to D
e
∀f, g ∈ D
Γ(f, g) = Γ(ϕ f, ψ g) on {ϕ = 1} ∩ {ψ = 1}
for any ϕ, ψ ∈ D. We then set
e ∞ = {f ∈ D
e : Γ(f ) ∈ L∞ }
D
where, as previously, Γ(f ) is put for Γ(f, f ).
e ∞ , D∞ ⊂ D
e ∞ and, if 1 ∈ D, D∞ = D
e ∞.
Proposition 3.4 1 ∈ D
e ∞ , equipped with the seminorm kf k0 = kΓ(f )1/2 kL∞
Proposition 3.5 The space F = D
satisfies the assumptions of Proposition 1.3.
We now define again the intrinsic measurable pseudometric ρ by
∀A, B ∈ Ω
e ∞ and Γ(f ) ≤ 1}.
ρ(A, B) = sup{ρf (A, B) : f ∈ D
Using again Proposition 1.3 and Proposition 2.4, we obtain:
e ∞ and, for any f ∈ D
e ∞,
Corollary 3.6 ρ is a measurable pseudometric, Lip∞ (ρ) = D
Lρ (f ) = kΓ(f )1/2 kL∞ .
3.3
Classical intrinsic metric
Consider again the situation of the previous subsection and suppose moreover that X is a
topological space and that µ is an inner regular Borel σ-finite measure. We set
Φ = {u ∈ D ∩ C : Γ(u) ≤ 1}
where C denotes the set of real continuous functions on X. As in Subsection 1.1, we define
the pointwise semimetric
ρΦ
0 (x, y) = sup{|u(x) − u(y)| : u ∈ Φ}.
This is the usual intrinsic semimetric (see for instance [4, 5]). We also may consider the
measurable pseudometric ρΦ associated to ρΦ
0 in the sense of Subsection 1.1. It is an open
Φ
problem in general, to compare ρ with the intrinsic measurable pseudometric ρ defined in
the previous subsection 3.2. K.-Th. Sturm gives in [25, 26] conditions for the equality in the
case of classical regular Dirichlet forms (on a locally compact space). In the next section, we
also study this problem in the case of Wiener spaces.
8
4
Abstract Wiener spaces
We consider in this section the particular setting of an abstract Wiener space (X, H, µ) in the
sense of L. Gross ([13]). We recall that, in particular, X is a separable Banach space, µ is
a centered Gaussian probability measure on X with support X, and (H, | |H ) is a Hilbert
space compactly embedded in X, which is called the Cameron-Martin space. We refer to [3]
for a precise definition and for the notation we adopt here. Actually, we could take the more
general framework considered in [8].
As shown in [3], there is a canonical Dirichlet structure (X, B, µ, D, E), where B denotes the
Borel σ-algebra of X, which is local and admits a carré du champ Γ. Moreover, 1 ∈ D and if
we set
D0 = {f (l0 , · · · , ln ) : n ∈ N, l0 , · · · , ln ∈ X 0 and f ∈ Cb∞ (Rn+1 )},
where Cb∞ (Rn+1 ) denotes the set of C ∞ -functions on Rn+1 which are bounded as well as all
their derivatives, then D0 ⊂ D∞ and D0 is dense in L2 . As a consequence, D∞ is weak*-dense
in L∞ .
The results of Subsection 3.1 are available and we can define the intrinsic measurable metric:
ρ(A, B) = sup{ρf (A, B) : f ∈ D∞ and Γ(f ) ≤ 1}
= sup{ρf (A, B) : f ∈ D and Γ(f ) ≤ 1}.
According to Corollary 3.2 and Corollary 3.3,
Lip∞ (ρ) = D∞ , Lip2 (ρ) = {f ∈ D : Γ(f ) ∈ L∞ },
and for any f ∈ D such that Γ(f ) ∈ L∞ , Lρ (f ) = kΓ(f )1/2 kL∞ .
4.1
µ-a.e. H-Lipschitz continuous functions
In [6], O. Enchev and D.W. Stroock introduced another notion of Lipschitz function (see also
[19, 20]):
Definition 4.1 Let f be a µ-class of real Borel functions. Then f is said µ-a.e. H-Lipschitz
continuous if there is a representative f˜ of f and a real constant C such that
∀x ∈ X ∀h ∈ H |f˜(x + h) − f˜(x)| ≤ C |h|H .
In this case, the H-Lipschitz constant of f is defined by
1 ˜
LH (f ) = ess sup{ sup
|f (x + h) − f˜(x)| : x ∈ X}.
h∈H\{0} |h|H
The main result of [6] is the following:
Theorem 4.2 Let f ∈ L2 . Then f is µ-a.e. H-Lipschitz continuous if and only if f ∈ D and
Γ(f ) ∈ L∞ .
In this case, LH (f ) = kΓ(f )1/2 kL∞ .
A direct consequence of the above theorem and Corollary 3.3 is the following:
Corollary 4.3
Lip2 (ρ) = {f ∈ L2 : f is µ-a.e. H-Lipschitz continuous}
and, if f ∈ Lip2 (ρ), then Lρ (f ) = LH (f ).
9
4.2
H-metric
We shall now give an equivalent definition of the intrinsic metric in terms of the CameronMartin space. We set, for x, y ∈ X,
ρH (x, y)
= |x − y|H
=
+∞
if x − y ∈ H
otherwise
and, for B ⊂ X, ρB
H (x) = inf{ρH (x, y) : y ∈ B}.
This metric was first considered by S. Kusuoka ([20]). In [7], S. Fang proved (see also [9]) the
following result.
Proposition 4.4 Let B be a Kσ (i.e. a countable union of compact sets) of X, with positive
2
measure. Then ρB
H ∈L .
Moreover, we have the following easy result:
Proposition 4.5 Let B be as in the previous proposition. Then ρB
H is µ-a.e. H-Lipschitz
continuous and LH (ρB
)
≤
1.
H
We now define, like in Subsection 1.1, for A, B ∈ Ω,
ρH (A, B) = inf{ρH (x, y) : x ∈ A, y ∈ B},
and
ρH (A, B) = sup{ρH (A0 , B 0 ) : A0 ∼ A, B 0 ∼ B}.
By the fact that, for any B a Kσ of positive measure, ρB
H is a Borel function, and by Proposition
H
1.2, ρ is a measurable pseudometric. Actually, µ is a Borel probability measure on the Polish
space X and ρH is lower semicontinuous on X×X. Therefore, the remark following Proposition
1.2 applies. Using Theorem 4.2 and Propositions 4.4 and 4.5 above, one can show the following
result.
Theorem 4.6 ([16, Theorem 4.6.]) For any A, B ∈ Ω,
ρH (A, B) = ρ(A, B).
Notice that in view of this theorem, Corollary 4.3 appears as a particular case of Theorem
2.14.
We deduce directly from Corollary 4.3 and Theorem 4.6 the following Corollary.
Corollary 4.7 Let f ∈ L2 . Then f is µ-a.e. H-Lipschitz continuous if and only if there
exists a constant C ≥ 0 such that
∀A, B ∈ Ω
ρf (A, B) ≤ C ρH (A, B).
Then,
LH (f ) = sup{ρf (A, B)/ρH (A, B) : A, B ∈ Ω and ρH (A, B) > 0}.
10
Finally, the next proposition states that the H-metric ρH is also the pointwise intrinsic
semimetric, in the sense of Subsection 3.3, associated with the Dirichlet space D. Therefore
in this case, the intrinsic measurable metric ρ also is the measurable pseudometric associated
with the pointwise intrinsic semimetric.
Proposition 4.8 ([16, Proposition 4.8.]) For every x, y ∈ X,
ρH (x, y) = sup{|u(x) − u(y)| : u ∈ D ∩ C and Γ(u) ≤ 1},
where C denotes the space of continuous functions on X.
5
Gaussian estimates
We consider in this section the general framework and the notation of Subsection 3.2. In
particular, ρ denotes the intrinsic measurable pseudometric associated to a local Dirichlet
form E with domain D and admitting a carré du champ operator Γ. There exists an L2 strongly continuous semigroup (Pt )t≥0 consisting of sub-Markovian symmetric operators such
that
1
∀f ∈ D
E(f ) = lim ((I − Pt )f, f )L2 .
t→0 t
We denote by λ the bottom of the spectrum of − A, where A is the infinitesimal generator of
(Pt )t≥0 :
E(u)
λ = inf
: u ∈ D, u 6= 0 .
kuk2L2
We first give integrated upper Gaussian estimates. The method to obtain such estimates
is wellknown and is probably due to M.P. Gaffney ([12]). In our general situation, we refer to
[26, Theorem 1.2] (see also [15]). Actually, in [26] as well as in [15], the existence of a carré
du champ operator is not assumed. The proof of Theorem 5.2 below is based on the following
proposition.
e ∞ . Assume Γ(ψ) ≤ 2γ. Then, for every t ≥ 0,
Proposition 5.1 Let f ∈ L2 and ψ ∈ D
keψ Pt f kL2 ≤ e(γ−λ) t keψ f kL2 .
Theorem 5.2 Suppose that A and B belong to Ω and µ(A) < ∞, µ(B) < ∞. Then, for all
t > 0,
ρ(A, B)2
(Pt 1A , 1B )L2 ≤ (µ(A) µ(B))1/2 exp −
−λt .
2t
Corollary 5.3 Suppose that A and B belong to Ω and µ(A) < ∞, µ(B) < ∞. Then,
ρ(A, B) = +∞ =⇒ ∀t > 0
11
(Pt 1A , 1B )L2 = 0.
If in particular, µ is a probability measure and (Pt )t≥0 is ergodic (this is actually the case in
the situation of Section 4), then
lim (Pt 1A , 1B )L2 = µ(A) µ(B),
t→+∞
and hence, for any A, B ∈ Ω, ρ(A, B) < +∞. Proposition 2.9 may then apply.
We now assume that µ is a probability measure and 1 ∈ D. The Gaussian behavior in small
time of the semigroup (Pt )t≥0 may be precisely described in terms of the intrinsic measurable
pseudometric ρ and of the distance functions ρA associated with ρ:
Theorem 5.4 ([15, Theorem 1.1]) For any A and B in Ω,
lim{2 t log[(Pt 1A , 1B )L2 ]} = −ρ(A, B)2 .
t→0
Theorem 5.5 ([15, Theorem 1.3]) Let A ∈ Ω. For any t > 0,
{Pt 1A = 0} ∼ {ρA = +∞}.
Moreover, 2 t log(Pt 1A ) converges in probability to − ρ2A on the set {ρA < +∞}.
6
Gaussian Concentration
We assume, in this section, that µ is a probability measure on X. As before, ρ denotes a
measurable pseudometric on (X, µ). The results presented in this section are essentially taken
from [17].
6.1
Concentration function
We begin with a definition.
Definition 6.1 For A ∈ Ω and r > 0, we denote by Ar the class (with respect to the relation
∼) {ρA < r}. We then define the concentration fonction α (related to the triplet (X, µ, ρ))
by
∀r > 0 α(r) = 1 − inf{µ(Ar ) : A ∈ Ω, µ(A) ≥ 1/2}.
Consider first the particular case described in Subsection 1.1 : The measurable pseudometric ρ is supposed to be associated with a pointwise semimetric ρ0 satisfying the hypothesis of
Proposition1.2. For A ⊂ X and r > 0, we set A0r = {ρA
0 < r}. We then define the “classical”
concentration function α0 (related to (X, µ, ρ0 )) by
∀r > 0
α0 (r) = 1 − inf{µ∗ (A0r ) : A ∈ Ω, µ(A) ≥ 1/2},
where µ∗ denotes, for example, the interior measure. The next proposition shows that this
classical concentration function coincides with the concentration function in the sense of Definition 6.1.
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Proposition 6.2 If α is the concentration function related to (X, µ, ρ), where ρ is the measurable pseudometric associated with ρ0 , then α0 = α.
We now come back to the general situation. The following proposition is the analogue, in
the general setting, of a classical result (see for example [1, Proposition 7.2.5]).
Proposition 6.3 We have
α(r) = sup{µ(F ≥ mF + r) : F ∈ Lip(ρ), Lρ (F ) ≤ 1, mF median of F }
and we may replace above Lip(ρ) by Lip∞ (ρ).
We finish this subsection with the definition of the Gaussian concentration property.
Definition 6.4 Let c be a positive constant. We say that the triplet (X, µ, ρ) satisfies the
Gaussian concentration property denoted by G(c) if the associated concentration function α
satisfies
r2
∃C ≥ 0 ∀r > 0
α(r) ≤ C exp −
.
c
We see by Proposition 6.2 that, in the particular case where the measurable pseudometric ρ
is associated with a pointwise semimetric ρ0 satisfying the hypothesis of Proposition 1.2, the
above Gaussian concentration property coincides with the classical one related to (X, µ, ρ0 )
(see for example [1, Définition 7.2.3]).
6.2
Kantorovich metric and Gaussian concentration
We denote by Π(µ) the set of all probability measures on X which are absolutely continuous
with respect to µ. We first define, in this setting, the Kantorovich metric (see, for example,
[24, p.88]).
Definition 6.5 We define the Kantorovich metric κ on Π(µ) by
Z
Z
∀ξ, η ∈ Π(µ) κ(ξ, η) = sup{ f dξ − f dη : f ∈ Lip∞ (ρ), Lρ (f ) ≤ 1} ≤ +∞.
Generally, κ is not a true metric on Π(µ). We have
∀ξ, η ∈ Π(µ)
κ(ξ, η) = 0
=⇒
ξ=η
if and only if ρ is a measurable metric (Definition 2.2).
We now define transportation type inequalities (this terminology being taken from [1]).
We recall that, if ξ = ϕ dµ belongs to Π(µ), the entropy of ξ is defined by
Entµ (ξ) = Eµ (ϕ log ϕ)
where Eµ denotes the expectation with respect to µ.
13
Definition 6.6 Let c be a positive constant. We say that the triplet (X, µ, ρ) satisfies the
transportation inequality denoted by T (c) if
q
∀ξ ∈ Π(µ)
κ(ξ, µ) ≤ c Entµ (ξ).
We can now state in this framework a result of S.G. Bobkov and F. Götze ([2]). In fact, by
our definition of the transportation inequality from the Kantorovich metric, the proof of [2,
Theorem 3.1.] (see also [1, Théorème 8.3.2]) works without modification.
Proposition 6.7 The transportation inequality T (c) is equivalent to the following property:
For all ψ ∈ Lip∞ (ρ) such that Lρ (ψ) ≤ 1, one has
c
∀t ∈ R
Eµ (exp(t ψ)) ≤ exp
t2 + t Eµ (ψ) .
4
The following theorem then shows the equivalence between a transportation inequality
and the Gaussian concentration property. For the proof of the first part, Marton’s argument
([22, 2, 21, 1, ...]) can be used.
Theorem 6.8 Let c be a positive real number. If T (c) holds, then G(c0 ) holds for any c0 > c.
Conversely, if G(c) holds, then there exists c0 > 0 such that T (c0 ) holds.
6.3
Logarithmic Sobolev inequality and Gaussian concentration
In this subsection, we particularize the previous framework. We consider the situation of
Subsection 3.1, with the same notation.
We recall that, if f ∈ L2 , the entropy of f 2 is defined by
Entµ (f 2 ) = Eµ (f 2 log f 2 ) − Eµ (f 2 ) log Eµ (f 2 ).
Definition 6.9 Let c be a positive constant. We say that the Dirichlet form (X, µ, D, E)
satisfies the logarithmic Sobolev inequality denoted by LS(c) if
∀f ∈ D
Entµ (f 2 ) ≤ c E(f ).
An important example is that of the canonical Dirichlet form on a Wiener space (Section 4).
In this case, according to [14] the logarithmic Sobolev inequality LS(4) holds.
In the rest of this subsection, c denotes a positive real number.
Using, as in classical cases, the so-called argument of Herbst (see [21, 1]) together with
Proposition 6.7, we can prove that, in this general situation too, a logarithmic Sobolev inequality implies a transportation inequality.
Theorem 6.10 If the Dirichlet form (X, µ, D, E) satisfies the logarithmic Sobolev inequality
LS(c), then the triplet (X, µ, ρ) satisfies the transportation inequality T (c/2).
As a consequence of the above theorem and Theorem 6.8, we get :
14
Corollary 6.11 If the Dirichlet form (X, µ, D, E) satisfies the logarithmic Sobolev inequality
LS(c), then the triplet (X, µ, ρ) satisfies the Gaussian concentration inequality G(c0 ) for any
c0 > (c/2). More precisely, one has
"
r
r
2 #
c
2
c
r≥
r−
log 2 =⇒ α(r) ≤ exp −
log 2
,
2
c
2
where α denotes the concentration function related to (X, µ, ρ).
We also obtain, as in [21, 1], the following result.
Theorem 6.12 If the Dirichlet form (X, µ, D, E) satisfies the logarithmic Sobolev inequality
LS(c), then, for any F ∈ D such that Γ(F ) ≤ 1, one has
2
∀r > 0
µ(F ≥ Eµ (F ) + r) ≤ exp − r2 .
c
6.4
Wasserstein metrics
We suppose in this subsection that X is a Polish space and that µ is a Borel probability. We
consider a pointwise semimetric ρ0 which is assumed to be lower semicontinuous on X × X.
Then, we suppose that ρ is the associated measurable pseudometric, in the sense of Subsection
1.1. This framework contains that of Section 4 (Wiener spaces) with ρ0 = ρH . The results of
this subsection are given in [17] in the case of Wiener spaces, but their extension to the more
general case considered here is easy, using Theorem 2.14 instead of Theorem 4.2.
If ξ and η are probability measures on X, we denote by Σ(ξ, η) the set of all probability
measures on X × X with marginal distributions ξ and η.
Definition 6.13 For p ∈ [1, ∞[ we define the Lp -Wasserstein metric by
1/p
Z
Wp (ξ, η) = inf{ [ρ0 (x, y)]p dπ(x, y) : π ∈ Σ(ξ, η)}
.
We shall first extend to this situation the classical Kantorovich representation theorem
(see for example [24, Theorem 2.5.6], [27, Theorem 1.14]).
Theorem 6.14 For every ξ, η ∈ Π(µ),
W1 (ξ, η) = κ(ξ, η),
where κ denotes the Kantorovich metric defined in Definition 6.5.
As usual, the proof of this theorem uses the duality theorem [18, Theorem 2.6] (see also for
example [27, Theorem1.3]).
For p > 1, we have the following extension of [24, Corollary 2.5.2] (see also [23, Theorem
5.2.1]). We also mention the deep study [10, 11] of the L2 - Wasserstein metric on abstract
Wiener spaces related to transportation problems. In the following statement, if h ∈ Lip∞ (ρ),
h̃ denotes a bounded ρ0 -Lipschitz continuous representative of h (see Theorem 2.14).
15
Theorem 6.15 Let p > 1 and let U be a Kσ of X with µ(U ) = 1. For every ξ, η ∈ Π(µ),
R
R
[Wp (ξ, η)]p = sup{ f dξ − g dη : f ∈ Lip∞ (ρ), g ∈ Lip∞ (ρ) and
∀x ∈ X, ∀y ∈ U, f˜(x) − g̃(y) ≤ [ρ0 (x, y)]p }.
Notice that, in the above statement, we may assume moreover that f˜ (resp. g̃) is an u.s.c.
(resp. l.s.c.) function. This follows from the proof of [17, Theorem 6.3].
In the case of Wiener spaces we may assume that U = U + H (notation of Section 4). Then
by modifying g̃ on the complement of U , we may replace U by X in the above statement.
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