Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition
For a smooth measure on an infinite-dimensional space, a “successful-filtration” condition is introduced and the Markov uniqueness and Rademacher theorem for measures satisfying this condition are proved. Some sufficient conditions, such as the well-known Hoegh-Krohn condition, are also considered....
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| Zitieren: | Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition / A.M. Kulik // Український математичний журнал. — 2005. — Т. 57, № 2. — С. 170–186. — Бібліогр.: 19 назв. — англ. |
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irk-123456789-1655692020-02-15T01:26:20Z Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition Kulik, A.M. Статті For a smooth measure on an infinite-dimensional space, a “successful-filtration” condition is introduced and the Markov uniqueness and Rademacher theorem for measures satisfying this condition are proved. Some sufficient conditions, such as the well-known Hoegh-Krohn condition, are also considered. Examples demonstrating connections between these conditions and applications to convex measures are given. Для гладкої міри на нескінченновимірному просторі введено умову "успішної фільтрації" та доведено марковську єдиність і теорему Радемахера для мір, що задовольняють цю умову. Розглянуто деякі достатні умови, такі як відома умова Хєег-Крона, наведено приклади, що демонструють зв'язок між цими умовами, та застосування до опуклих мір. 2005 Article Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition / A.M. Kulik // Український математичний журнал. — 2005. — Т. 57, № 2. — С. 170–186. — Бібліогр.: 19 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165569 519.21 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Kulik, A.M. Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition Український математичний журнал |
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For a smooth measure on an infinite-dimensional space, a “successful-filtration” condition is introduced and the Markov uniqueness and Rademacher theorem for measures satisfying this condition are proved. Some sufficient conditions, such as the well-known Hoegh-Krohn condition, are also considered. Examples demonstrating connections between these conditions and applications to convex measures are given. |
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Kulik, A.M. |
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Kulik, A.M. |
| title |
Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition |
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Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition |
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Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition |
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Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition |
| title_full_unstemmed |
Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition |
| title_sort |
markov uniqueness and rademacher theorem for smooth measures on an infinite-dimensional space under successful-filtration condition |
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Інститут математики НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/165569 |
| citation_txt |
Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition / A.M. Kulik // Український математичний журнал. — 2005. — Т. 57, № 2. — С. 170–186. — Бібліогр.: 19 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT kulikam markovuniquenessandrademachertheoremforsmoothmeasuresonaninfinitedimensionalspaceundersuccessfulfiltrationcondition |
| first_indexed |
2025-07-14T18:58:05Z |
| last_indexed |
2025-07-14T18:58:05Z |
| _version_ |
1837649878107291648 |
| fulltext |
UDC 519.21
A. M. Kulik (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
MARKOV UNIQUENESS AND RADEMACHER THEOREM
FOR SMOOTH MEASURES ON INFINITE-DIMENSIONAL
SPACE UNDER SUCCESSFUL FILTRATION CONDITION*
MARKOVS|KA {DYNIST| TA TEOREMA RADEMAXERA
DLQ HLADKYX MIR NA NESKINÇENNOVYMIRNOMU
PROSTORI ZA UMOVY USPIÍNO} FIL|TRACI}
For a smooth measure on an infinite-dimensional space, the successful filtration condition is introduced
and Markov uniqueness and Rademacher theorem for measures satisfying such condition are proved.
Some sufficient conditions, such as well-known Höegh-Krohn condition, are also considered. Some
examples demonstrating connections between these conditions and applications to convex measures are
given.
Dlq hladko] miry na neskinçennovymirnomu prostori vvedeno umovu „uspißno] fil\traci]” ta do-
vedeno markovs\ku [dynist\ i teoremu Rademaxera dlq mir, wo zadovol\nqgt\ cg umovu. Rozhlq-
nuto deqki dostatni umovy, taki qk vidoma umova X[eh-Krona, navedeno pryklady, wo demonstru-
gt\ zv’qzok miΩ cymy umovamy, ta zastosuvannq do opuklyx mir.
Introduction. The aim of this paper is to clarify some questions connected with the
notion of stochastic derivative (Sobolev derivative) on an infinite-limensional space
with a smooth measure. There are different closely related approaches to define
stochastic derivative (see for example [1], Ch. 7, items A.-E.) and the natural question
is whether definitions given by these approaches are equivalent. It appears that such an
equivalence, say, for most interesting case of W -derivative and G-derivative (exact
definitions will be given below), is nontrivial and requires the initial measure to satisfy
some structural conditions. The most known is so-called Höegh-Krohn condition (see
Definition 2.1 below), which can be regarded as a demand on the measure „to be close
to the product-measure”.
In this paper, we show that such condition is (is some situations) too restrictive and
can be replaced by another one, which we call „successful filtration” condition. We
also give class of the measures satisfying „successful filtration” property, for which
Höegh-Krohn condition may fail. We show that, under „successful filtration”
condition, equivalence of W- and G -derivatives (so-called „Markov uniqueness”
property, see next section for a deteiled discussion) holds true. Under the same
condition, we also prove the analogue of Rademacher theorem.
The structure of the paper is following. In Section 1 we give main definitions and
prove the description of G-derivative in terms of direction-wise Sobolev derivatives.
In Section 2, we prove Markov uniqueness property under the „successful filtration”
condition and give some sufficient conditions. In Section 3, the analogue of
Rademacher theorem is proved. In Section 4, we give some examples demonstrating
connections between different conditions sufficient for „successful filtration” property.
1. Stochastic derivatives on a space with a smooth measure. Let the separable
Banach space X and the probability measure µ on the Borel σ-algebra B ( X ) be
fixed. We suppose that there exists a separable Hilbert space H, densely embedded
into X by operator j ∈ L ( H, X ) , such that the measure µ is logarithmically
differentiable in every direction from j H. This means, by the definition, that there
exists the linear map (generalized random element)
ρ : H � h � ( ρ, h ) ∈ L1( X, µ )
such that for every function f from the set C X0, ( )cyl
∞ of the functions of the type
* This article was supported (in part) by Ministry of Education and Science of Ukraine (project
# 01.01/103).
© A. M. KULIK, 2005
170 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
MARKOV UNIQUENESS AND RADEMACHER THEOREM … 171
f F x xn( ) , , , ,* *⋅ = ⋅ ⋅( )1 … ,
(1.1)
F C n∈ ∞
0 ( )R , x x Xn1
* * *, ,… ∈ , n ≥ 1
(smooth cylindrical functions) the following integration by parts formula holds true:
E( ∇H f, h )H = – E f ( ρ, h ) , h ∈ H.
Here and below we use probabilistic notation E f f d
X
≡ ∫ µ , ∇H is the Gatoux
derivative w. r. t. H, which is defined for the functions of the type (1.1) as
∇ ⋅ = ′ ⋅ ⋅( )
=
∑H k
k
n
n kf F x x j x( ) , , , ,* * * *
1
1 … .
Throughout the paper we suppose that ρ has all moments, which means that
( , ) ( , )ρ µh L Xp p∈ ≥∩ 1 , h ∈ H. This implies due to the Banach theorem that for every
p ≥ 1 there exists cp < + ∞ such that
( , ) ( , )ρ µh c hL X p Hp
≤ , h ∈ H.
Consider operators ∇H and ∇h : f � ( ∇H f, h )H as the unbounded densely defined
operators
∇ ⊃ →∞
H p pL X C X L X H: ( , ) ( ) ( , , ),µ µ0 cyl ,
∇ ⊃ →∞
h p pL X C X L X: ( , ) ( ) ( , ),µ µ0 cyl , p ≥ 1.
It follows from the integration by parts formula that for every p ≥ 1 the adjoint
operators to ∇H , ∇h have domains, which separate points in L X Hp( , , )µ and
L Xp( , )µ respectively. Further we will need the explicit form of ( ∇H )* on a specific
set of functionals. Let us denote by C X H0, ( , )cyl
∞ the set of the elements of the type
g f hk k
k
n
=
=
∑
1
, f C Xk ∈ ∞
0, ( )cyl , hk ∈ H, k = 1 , … , n, n ≥ 1, (1.2)
due to the integration by parts formula the action of the adjoint operator to ∇H on the
element of the type (1.2) is given by
( ) ( , ) ( , )*∇ = − − ∇[ ]
=
∑H k k H k k H
k
n
g h f f hρ
1
.
Definition 1.1. Let p ≥ 1 be fixed, then
a) the closure Dp of ∇H in Lp sense is called Sobolev stochastic derivative
(W-derivative);
b) the adjoint operator Dp
G to the operator ( )*
( , ),
∇ ∞H C X H0 cyl
is called
generalized stochastic derivative (G-derivative);
c) the closure D p, h of ∇h in Lp is called stochastic derivative in direction h.
Domains of the corresponding operators are denoted by W Xp
1( , )µ , G Xp
1 ( , )µ ,
W Xp h, ( , )1 µ .
The following theorem shows that the notions of generalized derivative and
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
172 A. M. KULIK
derivative in direction h are closely connected.
Theorem 1.1. The function f belongs to G Xp
1 ( , )µ if and only if it belongs to
∩h H p hW X∈ , ( , )1 µ and there exists an element g L X Hf p∈ ( , , )µ such that D fp h, =
= ( gf , h )H a. s., h ∈ H. If so, g D ff p
G= .
Remarks. 1.1. One can see from the proof, that for a given p the integrability
condition on logarithmic derivative can be weakened to the following one:
( , ) ( , )ρ µh L L Xp q∈[ ]∩ , h ∈ H,
1 1
1
p q
+ = .
This condition is close to a necessary one, namely for every q < 2 one can give an
example of one-dimensional measure µ with logarithmic derivative ρ µ∈Lp( , )R \
Lq( , )R µ such that W Gp p
1 1( , ) ( , )R Rµ µ≠ (such an example the author first knew in a
private conversation with A. Yu. Pilipenko).
1.2. For the case p = 2 the statement of the theorem can be obtained using the
Dirichlet forms technique, namely it follows from Theorem 5.1 [2] (see also
Proposition A.1 from the Appendix there for the case p > 2). However, for p ≠ 2
(especially for p < 2) it is difficult to apply such technique because the question
about connections between W-, G -derivatives and derivative defined via Dirichlet
forms approach is complicated (see [1], Ch. 7). Therefore we suppose that the
straightforward proof given below is useful.
The proof of Theorem 1.1 is based on a stratification technique. Let h ∈ H be
fixed, let us choose some representation of X in the form of direct sum Y + 〈 jh〉,
where Y ⊂ X is some subspace. Take isomorphism X � x = y + tjh � ( y, t ) ∈ Y × R,
let π : X → Y be natural projection and µY is the image of µ under this projection.
It is well known that there exists a family {µy , y ∈ Y} of measures on R such that
for every A ∈ B ( X ) the function y t y t Ay� µ | ( , ) ∈{ }( ) is measurable and
µ µ µ( ) ( , ) ( )|A t y t A dyy
X
Y= ∈{ }( )∫ .
For every function f on X ≡ Y × R denote fy ( · ) = f (( y, · )) .
Lemma 1.1. The measure µ has logarithmic derivative ( ρ, h ) ∈ Lq ( X, µ ) i f
and only if for µY-almost all y ∈ Y has logarithmic derivative (w. r. t. usual
differentiation in R) ρy
h and
ρ µ µy
h q
y
Y
Y
t dt dy( ) ( ) ( )
R
∫∫ < + ∞.
In this case ( , )( , ) ( )ρ ρh y t ty
h= for µ-almost all ( y, t ) ∈ X.
For the proof of this lemma see [3].
Denote by G Xp h, ( , )1 µ the domain of the operator Dp h
G
, , adjoint in Lp ( X, µ ) to
operator
I C Xh : ( , ),0 cyl
∞ µ � f � ( ∇H )*( f h ) = – ( ρ, h ) f – ( ∇H f, h )H .
Due to Lemma 1.1 for µY-almost all y ∈ Y families Wp y
1
R, µ( ) and Gp y
1
R, µ( ) can
be defined analogously to the families W Xp h, ( , )1 µ and G Xp h, ( , )1 µ , denote
corresponding analogues of operators Dp, h , Dp h
G
, by Dp
y , Dp
G y, .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
MARKOV UNIQUENESS AND RADEMACHER THEOREM … 173
Lemma 1.2. 1. The function f belongs to W Xp h, ( , )1 µ if and only if for µ Y-
almost all y ∈ Y f Wy p y∈ ( )1
R, µ and
D f t dt dyp
y
y
p
y
Y
Y
[ ] < + ∞∫∫ ( ) ( ) ( )µ µ
R
.
In this case
D f y t D f tp h p
y
y, ( , ) ( )= [ ]
for µ-almost all ( y, t ) ∈ X.
2. The function f belongs to G Xp h, ( , )1 µ if and only if for µ Y-almost all
y ∈ Y f Gy p y∈ ( )1
R, µ and
D f t dt dyp
G y
y
p
y
Y
Y
, ( ) ( ) ( )[ ] < + ∞∫∫ µ µ
R
.
In this case
D f y t D f tp h
G
p
G y
y,
,( , ) ( )= [ ]
for µ-almost all ( y, t ) ∈ X.
Although the statement of the lemma is one of the main parts of the proof of
Theorem 1.1, we omit its proof because it is quite standard. The proof of statement 2
can be given analogously to the proof of Lemma 1.1 (see [3]). The proof of statement
1 can be obtained using the arguments analogous to given in [4].
Proof of Theorem 1.1. Due to Lemma 1.2 it is enough to prove statement of the
theorem when X = R and µ has logarithmic derivative
ρ µ∈[ ]L Lp q∩ ( , )R ,
1 1
1
p q
+ = .
First let us note that for the Lebesgue measure λ1 classes Wp
1 1
R, λ( ) and Gp
1 1
R, λ( )
(in our previous notations) coincide, one can easily obtain this by taking convolutions
of functions from Gp
1 1
R, λ( ) with smooth kernels. Next, note that µ has the density
α such that α λ1 1 1/ ,q
qW∈ ( )R and α ρα1 11/ /( / )q qq[ ]′ = , this is a corollary of the
previous observation and definition of logarithmic derivative. In particular, α is an
absolutely continuous function. Analogously, let f Gp∈ ( )1
R, µ , then the function
f f p= ⋅ α1/ belongs to Wp
1 1
R, λ( ) and its (Sobolev) derivative is equal D fp
G p⋅ α1/ –
– ( / ) /1 1p f pρα . Thus f is absolutely continuous on the open set {α > 0} . Let us
represent this set as a family of nonintersecting open intervals {( ak , bk ) , k ∈ J} and
show that
f Wa b pk k
÷( , ) ,∈ ( )1
R µ and
D f D fp a b a b pk k k k
1 1
÷ ÷( , ) ( , )[ ] = .
Let us study the case | ak | , | bk | < + ∞ (the case of unbounded interval can be
regarded analogously). Let us fix ε > 0 and choose δ = δ ( ε ) > 0 such that
+
+[ ] <
−
+
∫∫
b
b
a
a
p
p
p
k
k
k
k
f x D f x dx
δ
δ
µ ε( ) ˜ ( ) ( )
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
174 A. M. KULIK
and δ < ( bk – ak ) / 2. Choose the function ϕε ∈ ∞Cb ( )R such that supp ϕε ⊂ ( ak + δ,
bk – δ ) and
ϕε δ δ µ
− + − ( )
˜
( , ) ( , ),
D fp a b L a bk k p k k
÷ +
+ ϕ εε δ δ δ δ λ
− <+ − + −( )
˜
( , ) ( , ),
D fp a b L a bk k p k k
÷ 1 .
Then the function
f x f
a b
u u duk k
a b
x
a b
k k
k kε εϕ( ) ( ) ( )
( )/
( , )= +
+
+
∫2
2
÷
belongs to Cb
∞( )R and
f x f x f x D f x dxp
a
b
p
G p
k
k
ε ε µ( ) ( ) ( ) ( ) ( )− + ′ − →∫ 0 , ε → 0 +.
Therefore it is enough to prove that ÷( , ) ,a b pk k
W∈ ( )1
R µ and Dp a bk k
1 0÷( , ) = . Let us
note that due to the integrability condition on ρ we can take in the previous
considerations f = α . In this case, as far as ′ =∫ α ( )x dx
a
b
k
k 0 , f bkε ( ) → 0,
f akε ( ) → 0, ε → 0 +.
Thus the function ψ1 = α ·
÷( , )a bk k
belongs to the class Wp
1
R, µ( ) . Define
βm n
m
m
t
n
t
n
t t
n
,
/
/
( )
, , ,
, .
=
∈
≥
−1
0
1
1
1 1
1
Functions βm, n are globally Lipschitz, and therefore β ψ µm n pW, ( ) ,1
1∈ ( )R .
Moreover, one can easily verify that if n → ∞ then
β ψ ψ αm n m
m
a bk k,
/
( , )( )1
1→ ≡ ⋅÷
and
β ψ ψ ρψm n m mm, ( )1
1[ ]′ → ′ ≡
in Lp ( X, µ ) . This implies that ψ µm pW∈ ( )1
R, . Since
ϕm a bk k
→÷( , ) and ′ →ϕm 0
in Lp ( R, µ ) , m → ∞, ÷( , ) ,a b pk k
W∈ ( )1
R µ and Dp a bk k
1 0÷( , ) = .
The theorem is proved.
Connection between W- and G-derivative is more complicated.
Definition 1.2. Measure µ has the Markov uniqueness property of the order
p ∈ (1; + ∞) (notation µ ∈ MUp ) if W X G Xp p
1 1( , ) ( , )µ µ= .
Let us explain briefly the origin of the terminology. Let
�0( f, g ) = E( ∇H
f, ∇H g )H , f g C X, ( ),∈ ∞
0 cyl .
Except the fact that the map �0 ( ·, · ) is not closed, it satisfies all properties of a
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
MARKOV UNIQUENESS AND RADEMACHER THEOREM … 175
Dirichlet form (we don’t give here the corresponding definitions, referring the reader if
necessary to the book [5]). One can easily see that �0 is closable and its closure
coincides with the form
�( f, g ) = E( D2 f, D2 g )H , f g W X, ( , )∈ 2
1 µ .
At the same time, if G X2
1( , )µ is strictly larger than W X2
1( , )µ (obviously
W X2
1( , )µ ⊂ G X2
1( , )µ ) , then there exists one more closed extension on �0 , namely
�+ = ( )( , ) ,f g E D f D gG G
H2 2 , f g G X, ( , )∈ 2
1 µ .
Existence of two Dirichlet forms � and �+, extending �0 , is equivalent to
existence of two different Markov semigroups on L2( X, µ ) , such that their generators,
restricted on the set C X0, ( )cyl
∞ , are both equal to the infinite-dimensional elliptic
operator
L hH H h
k
k h
k
k k0
2
1 1
= ∇ ∇ ≡ ∇ − ∇
=
∞
=
∞
∑ ∑* ( , )ρ ,
here {hk} is an orthonormal basis (ONB) in H.
For the objects, introduced above, the following relations hold true.
Proposition 1.1. The following properties are equivalent:
1) µ ∈ MU2;
2) � is the unique Dirichlet form, coinciding with �0 on C X0, ( )cyl
∞ ;
3) there exists only one Markov semigroup on L 2( X, µ ) such that the restriction
of its generator on C X0, ( )cyl
∞ is equal to L0 .
Properties 1 – 3 hold true if
4) operator L0 is essentially self-adjoint on C X0, ( )cyl
∞ .
For the proofs and detailed discussions of the related topics we send the reader to
[2, 6, 7] (see also the bibliography given there). In [6] the assertion 4 (which is called
„strong uniqueness” property) was proved using the parabolic criterium of self-
adjointness for Hilbert space X under some regularity conditions imposed on the
vector logarithmic derivative ˜ :ρ X X→ . Unfortunately, in some situations these
results are hardly applicable, for example in some questions connected with the theory
of random operators X is a space of compact operators and is not a Hilbert one.
Therefore the natural question is how to obtain conditions, sufficient for Markov
uniqueness, in the „inner” terms of the measure, for example in the terms of its
logarithmic derivative.
2. Approach to Markov uniqueness based on finite-dimensional filtration.
One of the natural ideas about characterization of the functions, belonging to
W Xp
1( , )µ or G Xp
1 ( , )µ (which, as we believe, is up to the works [8, 9]), is to take
some sequence of finite-dimensional projections in X and give differential properties
of f in the terms of the sequence of its correspondent projections (i. e., conditional
expectations). Unfortunately, it is hard to obtain on this way a criterium, analogous to
given in Lemma 1.2 in the terms of the stratification of the space. This is caused by the
fact that the operation of projection (conditional expectation) is connected with
derivative in a more complicated way than the operation of stratification. Let us
consider this connection in a more details.
Suppose x x Xn1
* * *, ,… ∈ are fixed, M j x j x Hn= ⊂* * * *, ,1 … . Denote by FM σ-
algebra generated by x xn1
* *, ,… , fM = EM f ≡ E [f | FM ] , f ∈ L1( X, µ ) . Denote also
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
176 A. M. KULIK
for ϕ, ψ ∈ L1( X, µ ) such that ϕψ ∈ L1( X, µ )
cov ( , )M M M ME E Eϕ ψ ϕ ϕ ψ ψ= −[ ] −[ ]
— conditional covariation of ϕ, ψ under condition FM .
Lemma 2.1. Let h ∈ H, f W Xp h∈ , ( , )1 µ , then for every p̃ p≤ fM ∈
∈ W Xp h˜, ( , )1 µ and
( , ) ( , ) cov , ( , )˜ , ˜D f h E D f h f hp h M H M p M= + ( )ρ . (2.1)
This result is well known (up to notations, in which we change the term E M f ( ρ,
h ) – E f E hM M ( )ρ, by covM ( f ( ρ, h )) , its proof can be found for instance in [1],
Proposition 7.1.9.
Consider the bilinear continuous map
R h g h gM
Mµ ρ ρ: ( , ) cov ( , ), ( , )� ( ), h, g ∈ H.
One can easily see that RM
µ is a weak random operator in H in a sense of Skorokhod
(see [10]). Consider operator R̃ P R PM
M
M
Mµ µ= (PM is the projector on M in H ),
since space M is finite-dimensional, R̃M
µ is a bounded random operator in M.
Definition 2.1. The measure µ is said to have the successful filtration property
of the order p ∈ (1, + ∞] (notation µ ∈ SFp ) if there exists a sequence
x Xn
* *{ } ⊂ , separating points in X , such that for the corresponding sequence
M j x j xn n= * * * *, ,1 …
sup ˜
( ) ( , )/n
M
M L X
R n
n p
µ
µL
2
< + ∞ ,
here we denote f E fL
r r
r
= ( )1/
for all r > 0, i. e., ⋅ Lr
is not necessarily
a norm.
It was mentioned in [10] that it is difficult to give conditions on a weak random
operator, necessary and sufficient for this operator to be bounded. On the other hand,
one can give a criterium for the operator to be a random Hilbert – Schmidt operator
(see [10], Ch. 1, Sect. 1.2). Similarly, we can give a criterium on a sequence of
operators to have uniformly bounded (in Lp/2 sense) Hilbert – Schmidt norms.
Imposing these conditions on R̃Mn
µ , and then using Jensen’s inequality for conditional
expectation we obtain the well known condition, which sometimes is named Höegh-
Krohn condition.
Definition 2.2. The measure µ satisfies Höegh-Krohn condition of the order
p ∈ (1, + ∞] (notation µ ∈ HKp ) if there exists a sequence x Xn
* *{ } ⊂ , separating
points in X, such that the sequence e j xn n≡{ }* * is an ONB in H and
sup ( , ) ( , )
/ ( , )
n
k M k
k
n
L X
e E e
n
p
≥ =
−[ ] < + ∞∑
∨
1
2
1
2 1
ρ ρ
µ
.
Obviously, condition µ ∈ HKp is more strong than µ ∈ SFp (since condition on an
operator to be Hilbert – Schmidt is more strong than condition to be bounded), but it is
more easy to check. Let us give another sufficient condition for µ ∈ SFp
.
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MARKOV UNIQUENESS AND RADEMACHER THEOREM … 177
Definition 2.3. The measure µ belongs to the class Bp , p ∈ (1, + ∞] , if
( , ) ( , )ρ µh G Xp∈ 1 , h ∈ H and random operator Bµ in H, given by
B h D hp
G
µ ρ= − ( , ), h ∈ H,
is bounded and
B
H p
µ L ( ) /2 1∨
< + ∞ .
Let us note that operator Bµ is known as an inner characteristic of the measure, in
terms of which some analytical properties can be described in a natural and compact
way. Say, if Bµ ≥ cIH , c > 0, then the well known Bakry – Emery criterium gives
log-Sobolev inequality for stochastic derivative [6], under the same condition the
analogue of Clark – Ocone representation theorem hold true [11], the same operator is
involved into the formula for the Jacobian for nonlinear transformations of measure µ
[12]. Further we give a sufficient condition for µ ∈ SFp in the terms of this object.
For a fixed sequence xn
*{ } , such that e j xn n≡{ }* * is ONB in H, denote Xn =
= jMn , note that the measure µ µn Mn
= |F can be regarded as the projection of the
measure µ on Xn w. r. t. canonical decomposition X X x xn n= + 1
* *, ,…
�
. Every
measure µn , n ≥ 1, has logarithmic derivative ρn w. r. t. directions from Mn and
(see [3])
( , ) ( , )ρ ρn H Mh E h
n
= , h ∈ Mn , n ≥ 1. (2.2)
This allows us to define classes W Xp n n
1( , )µ , G Xp n n
1 ( , )µ , n ≥ 1.
Definition 2.4. The measure µ belongs to the class Bp
P
, p ∈ (1, + ∞] , if there
exists a sequence x Xn
* *{ } ⊂ , separating points in X , such that the sequence
e j xn n≡{ }* * is an ONB in H, for every n ≥ 1 µn ∈ Bp and
sup
( ) /n
M p
B
n n
µ L 2 1∨
< + ∞.
Lemma 2.1. If µ ∈Bp
P , then µ ∈ SFp .
Proof. Standard approximation arguments together with statement of Theorem 1.1
imply that if µ ∈Bp
P then µ ∈ Bp . Due to (2.1) and (2.2)
cov ( , ), ( , ) ( , ) ( , )M M H Mn n n n
h g E B h g B h gρ ρ µ µ( ) = − , h, g ∈ Mn , (2.3)
which together with Jensen’s inequality gives that
sup ˜
( ) ( , )/n
M
M L X
R n
n p
µ
µL
2
≤
≤
B B
H p n
M pn n
µ µL L( ) / ( ) /
sup
2 1 2 1∨ ∨
+ < + ∞ .
The lemma is proved.
Theorem 2.1. Let µ ∈ SFp , then µ ∈ MUp .
Remark 2.1. It follows from the previous discussion and the statement of the
theorem that if either µ ∈ HKp or µ ∈Bp
P , then µ ∈ MUp .
Proof of the theorem. We follow the scheme given in the proof of Theorem
7.1.11 [1] (except one point, which will be discussed later). Let sequence xn
*{ } from
Definition 2.1 be fixed, suppose for a while that we already know that
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178 A. M. KULIK
W X G Xp n n p n n
1 1( , ) ( , )µ µ= for every n ≥ 1. Let f G Xp∈ 1 ( , )µ be fixed, taking if
necessary ˜ ( )f f= ϕ , ϕ ∈ ∞Cb ( )R , we can suppose that f ≤ 1. Consider the
sequence f E fn Mn
={ }, f fn → in Lp ( X, µ ) . Due to Lemma 2.1 for every h ∈ H
D f E D f f P hp h
G
n M p P h
G
M Mn M n nn
, , cov , ( , )= + ( )ρ . (2.4)
The H-norm of the first term in the right-hand side of (2.4) is estimated by
sup sup, ,
h
M p P h
G
M
h
p P h
G
M p
G
H
H
n M n
H
M n
E D f E D f E D f
n n= =
≤ ≤
1 1
.
In order to estimate the second term we use inequalities
cov ( , ) cov ( , ) cov ( , )/ /
M M Mϕ ψ ϕ ϕ ψ ψ≤ [ ] ⋅[ ]1 2 1 2,
valid for ϕ, ψ ∈ L2( X, µ ) , and
cov ( , ), ( , ) ˜
( )M
M
M Hh h R hρ ρ µ( ) ≤ L
2 , h ∈ M.
This gives that
sup cov , ( , ) ˜
( )
/
h
M M
M
M
H
n n
n
n
f P h R
=
( ) ≤
1
1 2
ρ µ L
(note that cov ( , )Mn
f f ≤ 1 since | f | ≤ 1) , and consequently the sequence D fp
G
n
(which is equal to Dp fn due to supposition made at the beginning of the proof) is
bounded in Lp ( X, µ, H ) . Since the closed unit ball is weakly compact in Lp ( X, µ, H ) ,
we can take a subsequence {nk} such that D fp nk
weakly converges. From the
elements of this sequence we can compose the sequence of convex linear combinations,
strongly convergent to the same limit. Since the sequence of the same combinations
composed from the sequence fnk{ } converge to f in Lp ( X, µ ) , we obtain the needed
statement.
Therefore in order to prove the theorem we need Markov uniqueness property for
finite-dimensional projections. The question of finite-dimensional Markov uniqueness
is studied in details (see [7, 13] and references there), but most of the existing results
are hardly applicable in our situation, because they have some „structural” conditions
on the measure. Say, in [13] the density of the measure has to be positive a. s. with
respect to Lebesgue measure (note that the proof of implication µ ∈ HKp ⇒ µ ∈ MUp
in [1] refers to results of [13] and therefore formally is correct only for such measures
that their finite-dimensional projections have a. s. positive densities). Fortunately, we
have the following moment condition: logarithmic derivative of µ, and conseguently
logarithmic derivatives of all µn , have moments of every order. Let us show finite-
dimensional Markov uniqueness under this condition.
Let n ≥ 1 be fixed, then µn considered as a measure on R
n, has a density αn
(see [3]). Moreover (see [1], proof of Proposition 4.3.1) for every p ≥ 1
α λn
p
p
n nW1 1/ ( , )∈ R , and by Sobolev inclusion theorem we have that αn ∈ C ( R
n
) .
Now let f Gp
n
n∈ 1 ( , )R µ , then ϕ αn n
pf= 1/ belongs to W dp
n n1( , )R λ and its
derivative is equal to D f p fp
G
n
p
n n n
p⋅ +α ρ α1 11/ /( / ) , where ρn = ∇ ( ln αn ) is the
logarithmic derivative of µn . Let us take κ ∈ ∞C n
0 ( )R , κ ≥ 0, κ( )x dxn
R∫ = 1, and
put
ϕ
ε
ϕ κ
ε
ε
n n nx x y
y
dy
n
( ) ( )= +
∫1
R
,
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MARKOV UNIQUENESS AND RADEMACHER THEOREM … 179
r x x y
y
dyn n
p
n
ε
ε
α κ
ε
( ) ( )/= +
∫1 1
R
.
Due to standard properties of convolution ϕ ϕε
n n→ , ε → 0, in W dp
n n1( , )R λ and
rn n
pε α→ 1/ , ε → 0, uniformly on every compact. Multiplying, if necessary, f on a
function from C n
0
∞( )R , we can suppose that f has a compact support and therefore
supports of all functions ϕε
n belong to some compact K. Choose δ( ε ) such that
r x xn n
pε δ ε α( ) ( ) ( )/+ > 1 , x ∈ K, ε > 0.
It is easy to see that it can be chosen in such a way that δ( ε ) → 0, ε → 0 +. Now one
can verify that the family of the functions
f
r
n
n
n
ε
ε
ε
ϕ
δ ε
=
+ ( )
belongs to W dp
n
n
1( , )R µ and converge to f in W dp
n
n
1( , )R µ as ε → 0 +.
The theorem is proved.
3. Rademacher theorem under „successful filtration” condition. In this section
we are going to discuss the infinite-dimensional analogue of the following famous
theorem by Rademacher.
Proposition 3.1. For every n ∈ N, p ≥ 1 the following two properties for a
function f ∈ Lp ( R
n, λn
) are equivalent:
i) f has a λ n-modification, which is a globally Lipschitz function with Lipschitz
constant C;
ii) f Wp
n n∈ 1( , )R λ and D f Cp
n
d
R
≤ λ -a. s.
The natural question is how to extend this result to the situation described in
Section 1. In this case the Lipschitz property of the function has to be changed, since
the space of differentiability is H but not X.
Definition 3.1. The function f L Xp∈ ( , )µ is called to be H -Lipschitz with the
constant C if
f x jh f x C h H( ) ( )+ − ≤ , x ∈ X, h ∈ H.
Considering the Rademacher theorem as consisting of two parts „i) ⇒ ii)” and
„ii) ⇒ i)”, one can see that the first part is much easier to give an analogue.
Proposition 3.2. Let f L Xp∈ 1 ( , )µ and f has an H -Lipschitz modification
with the constant C. Then f G Xp∈ 1 ( , )µ and D f Cp
G
H
≤ µ-a. s.
This result was obtained by S. Kusuoka, see [14, 15]. One can easily prove it using
Proposition 3.1 for n = 1 and statement 2 of Lemma 1.2.
The proof of the analogue of the second part in the infinite-dimensional case is
more complicated and requires additional properties on the measure to hold true. It
was proved in [16] for Gaussian measure and in [17] for measure satisfying some more
strong version of condition µ ∈ HK∞ .
Our main result in this section is the following theorem
Theorem 3.1. Let µ ∈ SF∞ to satisfy the following condition: for every h ∈ H
there exists some ε = ε( h ) > 0 such that E hexp ( , )ε ρ{ } < + ∞.
Then for every f W Xp∈ 1( , )µ , such that D f Cp H
≤ a. s., there exists a
modification f̃ of f, which is H-Lipschitz with constant C.
Remark 3.1. Under condition of the theorem W X G Xp p
1 1( , ) ( , )µ µ= .
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180 A. M. KULIK
Proof of the theorem. Let us fix a sequence xn*{ } from the definition of the class
SF∞ , f E f fn Mn
= → , n → ∞ , a. s. We can suppose that | f | ≤ 1 a. s., which
implies, analogously to the proof of Theorem 2.1, that
sup
( , , )n p n L X H
D f C
∞
= < + ∞
µ 1 ,
a. s. Due to exponential integrability condition on the logarithmic derivative, every
projection µn of µ has a continuous positive density (see [1], Proposition 4.3.1), and
therefore we can apply finite-dimensional Rademacher theorem to every fn and obtain
a sequence of functions f̃n{ } such that f̃ fn n= a. s. and
˜ ( ) ˜ ( )f x jh f x C hn n H+ − ≤ 1 , x ∈ X, h ∈ H.
Denote
A x f x f xn n0 = ∃ ={ }→∞: lim ˜ ( ) ˜( ) .
Since measure µ is quasiinvariant w. r. t. H (see [1], Proposition 4.3.1), ∀ h ∈ H
µ( A0 + jh ) = 1. Let us choose in H a dense countable subset H0 and put
A A jh
h H
1 0
0
= +
∈
( )∩ .
Then µ( A1 ) = 1 and, by the construction,
∀ x ∈ A1 , h H f x jh f x jhn n∈ ∃ + = +→∞0 lim ˜ ( ) ˜( ).
The fact that the sequence f̃n{ } is uniformly H-Lipschitz implies that there exists the
limit lim ˜ ( )n nf x jh→∞ + for every x ∈ A1 , h ∈ H. Therefore the following definition
is correct:
ˆ( )
, ,
lim ˜ ( ), ,
f x
x A jH
f x x A jHn n
=
∉ +
∈ +
→∞
0 1
1
one can see that f̂ is a modification of f, which is H-Lipschitz with the constant
C1 . Now we need to „correct” the modification f̂ in order to make Lipschitz
constant equal to C . Let us show that for every Mn one can choose another
modification f̂n such that
ˆ ( ) ˆ ( )
, , ,
, , .
f x jh f x
C h x X h H
C h x X h M
n n
H
H n
+ − ≤
∈ ∈
∈ ∈
1
We choose a sequence {ϕε} ⊂ C( R
n
) , convergent to δ0 , and put
ˆ ( ) ˆ( ) ( ) ( )f x f x jh h dhn
n
Hn
ε
εε ϕ λ= +∫ , x ∈ X.
The family ˆ ,fn
ε ε >{ }0 is convergent a. s. to f̂ as ε → 0 and satisfy condition
ˆ ( ) ˆ ( )
, , ,
, , .
f x jh f x
C h x X h H
C h x X h M
n n
H
H n
ε ε+ − ≤
∈ ∈
∈ ∈
1
This gives, after the considerations analogous to given above, needed modification f̂n .
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MARKOV UNIQUENESS AND RADEMACHER THEOREM … 181
At last, taking n → ∞ and making again the same considerations, we obtain the
needed statement.
The theorem is proved.
Let us note that the exponential integrability condition, imposed in Theorem 3.1 on
the logarithmic derivative, is crucial: if it fails, then the statement of the Rademacher
theorem can fail even in the simplest one-dimensional case.
Example 3.1. Let X = H = R1 and µ has the density of the form
p x c
x
x( ) exp= − −
α α
α1
, x ∈ R,
α > 0 is some constant. Logarithmic derivative now is equal
ρ α α
α( ) ( )x x
x
x= −
+
−sign
1
1
1
and has all moments. However, the Rademacher theorem is not true: the function
f = +÷
R
belongs to Wp
1( , )R µ with Dp
f = 0 for every p ≥ 1 (see the proof of
Theorem 1.1), but it has not any continuous modification. Note that taking α large
enough we see that every condition of the type
E h rexp ( , ) /ε ρ 1{ } < + ∞ , r > 1, h ∈ H, ε = ε ( h ) > 0
is not sufficient for Rademacher theorem to hold true.
4. Examples. In this section we give some examples of measures satisfying
„successful filtration” condition. The first example is well known.
Example 4.1 (Gaussian measure). Let ( X, H, µ ) be a canonical Wiener space,
then for every M ⊂ H R̃M
µ = 0 and µ ∈ HK∞ . More general, if µ have a density
pµ from the class Gp
1 with respect to some (smooth) product measure ν and
D p
p
L X Hp
G
p
µ
µ
µ∈ ( , , ),
then µ ∈ HKp .
In the second example the condition µ ∈ SFp is much harder to be proved. It
shows, in particular, that classes Bp
P and Bp are in general different.
Example 4.2 (Stratified measure). Let µ be equal
µ µ µ= +1
2
1 2( ) ,
where µ1, 2 are Gaussian measures on X with covariation operators jj*, 4j j*
correspondingly.
One can show that there exist H-invariant sets A1, 2 such that A1 ∩ A2 = Ø, A1 ∪
∪ A2 = X and µ δi
j ijA( ) = , i, j = 1, 2. This gives that logarithmic derivative of µ
is equal ( ρ, h ) ( · ) = – ( ·, h ) · η, where
η = +÷ ÷A A1 2
1
4
and ( ·, h ) is the measurable linear functional on X, correspondent to h . We will
show that
1) µ ∈B HKp
P
p∩ , p ∈ (1, + ∞);
2) µ ∈ B∞ , µ ∉ ∞BP
, µ ∉ ∞HK .
First, note that η is H-invariant, and therefore Dp
Gη = 0, Bµh = ηIH , which
gives that µ ∈ B∞ .
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182 A. M. KULIK
It is a little bit easier to study the question whether µ ∈Bp
P , let us start with it.
For every fixed basis corresponding projection µn of the measure µ on Rn has the
density equal to
p x f
x
n n n
n( ) =
( )
1
2 2
2
π
,
where || · ||n is Euclid norm in Rn and
f u e en
u
n
u( ) /= +
− −1
2
1
2
4
(further we omit subscript at f ) .
Straightforward computations give that
B
f
x
f
x
x
f
x
f
x
f
x
f
xn nM
n
n
n
n
n
n
n
µ L ( )
−
′
≤
′′
−
′
2
2
2
2
2
2
2
2
2
2
2
2
2
2
. (4.1)
The functions ′( ) ⋅f fn n/ ( ) are uniformly bounded, and therefore the question whether
µ belongs to Bp
P is determined by the right-hand side of (4.1). One has
′′ − ′
=
+
− −
− −
f u
f u
f u
f u
e
e e
n
u u
u
n
u
( )
( )
( )
( )
/
/
2
4
4
2
9
16
1
2
1
2
, u ≥ 0. (4.2)
Since the function (4.2) at the point
u
n
n = 8
7
2ln
takes value
9
16
1
2
02 = >const
and un → ∞, n → ∞ , one has that ess sup
B
n nMµ ( )
( )
⋅ → ∞L , n → ∞ . This gives
that µ ∉ ∞BP . For p < + ∞
x
f
x
f
x
f
x
f
x
p x dxn
p
n
n
n
n
p
n
n
2
2
2
2
2
2
2
2
2
2
R
∫
′′
−
′
( ) =
=
9
16 2
1
2
1
2
2
8
8 8
2 1
2
2 2
( )
+
∫
−
− −
−
p
n
p
n
n
x
p
x
n
x
p
n
n
n n
x
e
e e
dx
R
π
=
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MARKOV UNIQUENESS AND RADEMACHER THEOREM … 183
=
9
16 2
1
2
2
1
2
2 5
16
3
16
3
16
2 1
2
2 2
( ) ( )
( ) +
∫
−
−
−
p
n
p
n n
x
n
x n x
p
n
n
n n
x
e
dx
e e
R
π
≤
≤
9
16
1
2
8
5
1
2
2 2
5 2
02 1
2
2
2
( )
( )
→−
−
∫
p
p
p
n
n
n
p
n
y
n
ny
e dy
R
π
, n → ∞,
since
R
n
ny
e dy O nn
p
n
y
p∫ ( )
=
−2
2
2
2
π
( ), n → ∞.
This gives that µ ∈Bp
P , p < + ∞.
In order to prove the Höegh-Krohn condition let us denote by rMn
the random
Mn-valued element such that r h hM Mn n
, ( , )( ) = ⋅ , h ∈ Mn . Then operator R̃Mn
µ can
be represented in the form
˜ cov ( , )R r rM
M M M
n
n n nµ η η= ⊗[ ] ,
and its norms in L ( Mn ) and L2( Mn ) are both equal to rM M Mn n n
2
cov ( , )η η . This
implies that in the case under consideration the property µ ∈ SFp , which is already
proved, implies µ ∈ HKp .
Note that in the considered example using „successful filtration” condition is not a
proper way to prove Markov uniqueness or Rademacher theorem. In this case one can
show that ÷A pi
W X∈ 1( , )µ with Dp Ai
÷ = 0, i = 1, 2, and the both mentioned results
can be proved separately on A1, 2 using result of Example 4.1. The given example is
interesting from the point of view of explicit computations, demonstrating connections
between classes of measures introduced in Section 2.
Example 4.3 (Convex measure). One can see from the previous example that the
straightforward verification of the condition µ ∈ SFp (or sufficient conditions µ ∈
∈ HKp , µ ∈Bp
P ) can be rather complicated since these conditions are imposed, in fact,
on an infinite family of projections on initial measure. It occurs that these difficulties
are greatly reduced in the case when µ is a convex measure.
Definition 4.1 [18]. The measure µ is said to be convex if for every compacts
A, B ⊂ X and every α ∈ [0, 1]
µ α α µ µα αA B A B+ −( ) ≥ −( ) ( ) ( )1 1 .
Lemma 4.1. Let µ be convex and satisfy assumptions of Section 1. Then µ ∈
∈ Bp
P iff µ ∈ Bp .
Proof. We have to prove only implication µ µ∈ ⇒ ∈B Bp p
P. Under conditions
of the lemma every finite-dimensional projection of µ has a density of the type
exp (– V ) , where V is a convex function (see [18] for a characterization of the class of
convex measures in the terms of finite-dimensional projections). Then for every
projection µn correspondent operator B
nµ (which is well defined due to supposition
µ ∈ Bp and Lemma 2.1) is positive, i. e., B h h
n nMµ ,( ) ≥ 0 a. s. for every h ∈ MP .
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184 A. M. KULIK
This is a corollary of the relation B V
nµ = ∇2 , which is to be understood in a
generalized sence, and can be obtained from the definition of stochastic derivative.
Then by (2.3) one has that
cov ( , ), ( , ) ( , )M M Hn n
h h E B h hρ ρ µ( ) ≤ , h ∈ Mn .
Now the needed statement holds true due to Cauchy inequality valid for covM .
The lemma is proved.
Let us give a concrete example of using of this result. Fix the equence xn*{ } (such
that e xn n={ }* is ONB in H) , and let us choose a symmetric operator A ∈ L ( H )
such that A ≥ γIH for some γ > 0. We suppose that
Ae ek j H
k j
,( ) = + ∞
≠
∑ 2
. (4.3)
Also for notational convenience we demand (this limitation can be removed) that
Ae j Xn ∈ * * , n ≥ 1.
Let us fix an even function ϕ ∈Cb
1( )R such that inf ( )
R
′ > −ϕ γx , and consider
measure µ, which have logarithmic derivative w. r. t. H, and on en this derivative is
given by the formula
( , ) , ,* *ρ ϕe x j Aen n n= − ⋅( ) − [ ] ⋅
−1
, n ≥ 1. (4.4)
We are not going to discuss the uniqueness problem since even for ϕ = 0 it is
nontrivial (see, for instance, [1], Ch. 6.4). We take by µ any measure satisfying
(4.4) and additional condition to be H-ergodic (or nonstratified):
D f fp = ⇔ =0 const , f W Xp∈ 1( , )µ .
Note that every two such measures (if exist) are mutually singular and every measure
satisfying (4.4) is a mixture of such measures.
The existence of H-ergodic measure satisfying (4.4) can be proved, for instance,
using some limit procedure. One possible choice of the limiting measures is the finite-
dimensional measures νn with the density ot the type exp −[ ]Vn with
∂
∂
ϕ
x
V x x A x
k
n k k
n
n( ) ( ) ˜ ,= − − ( )
R
, x ∈ R
n, k = 1, … , n,
where vectors ˜ , , ˜A An
n
n
1 … are given by equalities
˜ , ,*A x j Ae x jek
n
k r r
r
n
n( ) = [ ]−
=
∑
R
1
1
, x = ( x1 , … , xn ) ∈ R
n.
Another possible choice is to take the Gaussian measure κ 0 with covariation operator
equal jAj* and consider the sequence {κn} with
d
d
x C x xn
n k
k
nκ
κ
ψ
0 1
( ) exp , *= − ( )
=
∑ ,
x ∈ X, ψ ϕ( ) ( )⋅ =
⋅
∫ u du
0
.
By the construction there exist α > 0, β < + ∞ such that ν α βn MC
n
∈ ( , ) ,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
MARKOV UNIQUENESS AND RADEMACHER THEOREM … 185
κ α βn HC∈ ( , ), n ≥ 1 (by the definition, see [19], µ α β α∈ ⇔BC IH H( , ) ≤ Bµ ≤
≤ βIH a. s.). This implies (see [19]) that for every p > 1 there exist constants ap =
= a( p, α, β, H ) , bp = b( p, α, β, H ) such that
a j x x x y x dy dx b j xp H
p
X
n
X
p
n p H
p
* * * * * *, , ( ) ( )≤ − ≤∫ ∫ κ κ , (4.5)
x X* *∈ , p > 1, n ≥ 1.
The same estimate is valid also for νn for every x x xn
* * *, ,∈ 1 … . This implies, that
under additional condition that inclusion j is p-summating for some p, the sequences
{νn} , {κn} are weakly compact. Let us take by µ some limiting point of one of
these sequences. Since {νn} , {κn} are convex for every n, µ is also convex (see
[18]). By the same reason we have that µ ∈ CH( α, β ) (see [19]).
Due to Lemma 4.1, µ ∈ ∞BP , and therefore µ ∈ SF∞ . Let us consider the question
whether µ ∈ HKp . First let us consider the case when the sequence {xn} from the
Definition 2.2 is the same with the sequence used in the construction of µ. Let us
show that for every n ≥ 1 and y X* *∈ such that j y j x xn* * * * *, ,� 1 … ,
E y E y j yM Hn
⋅ − ⋅[ ]( ) ≥, ,* * * *
2 2
δ , (4.6)
where the constant δ > 0 depends only on α, β , H . We take decomposition X =
= jj y X* * ˜+ , where ˜ *X y=
�
and denote ˜ * *M j y=
�
. Now (4.6) follows
from inequalities
E y E y y y j y
M M M Hn n
⋅ − ⋅[ ]( ) ≥ ⋅ ⋅( ) ≥, , cov , , ,* *
˜
* * * *
2 2
δ ,
the last inequality is a corollary of (4.5) and the fact that almost all conditional
measures on x jj y+ * * belong to B
j y* *
( , )α β . Due to (4.3), (4.6) we have that
lim ( , ) ( , )
/ ( , )
n
k M k
k
n
L X
e E e
n
p
→∞ =
−[ ] = + ∞∑
∨
ρ ρ
µ
2
1
2 1
for every p ≥ 1.
The question whether condition of Definition 2.2 holds for some sequence ˜*xn{ } is
more difficult and rely on the form of the function ϕ . If ϕ( u ) = u2 (µ is Gaussian)
and A has eigenbasis, then one should take by ˜*xn{ } this basis. On the other hand, if
ϕ( u ) = ( γ / 2) sin u, then such sequence does not exist and µ ∉ HK2 . The proof of this
fact is analogous to the previous considerations (we omit details in order to shorten
exposition), with inequality (4.6) combined with inequality
inf sin( ) ( )
,a b
cu au b du K c+ +[ ] ≥∫ 2 2µ
R
, | c| ≤ 1, K = K( α, β ) > 0,
valid for every one-dimensional measure µ ∈ CR( α, β ) .
Let us end this example with a conclusive remark. Conditions (4.4) can be
interpreted in such a way that µ is an invariant measure for the following elliptic
operator
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
186 A. M. KULIK
L f x f x x Ax fH H0 ( ) ( ) ( ) ,= − − +[ ] ∇( )∆ Φ , f C X∈ ∞
0, ( )cyl , (4.7)
where generalized random elements Φ, Ax in H are given by formal series
Φ( ) , *x x x en n
n
= ( )∑ ϕ ,
Ax j Ae x en
n
n= [ ]−∑ * ,
1
, x ∈ X.
We can formulate the result of our considerations in the following form.
Proposition 4.1. Let M ( L0 ) be the family of the invariant measures for
operator (4.7), then M ( L0 ) is nonemply and for every µ ∈ M ( L0 ) both
Rademacher theorem and Markov uniqueness for every p > 1 hold true.
The statement of the proposition for µ ∈ M ( L0 ) , which is not H-ergodic, one can
prove by stratifying µ into a mixture of mutually singular H-ergodic measures and
then using previous results.
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Received 17.03.2004
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