Doc Math J DMV

Which Moments of a Logarithmic Derivative

Imply Quasi invariance

Michael Scheutzow Heinrich v Weizsacker

Received June

Communicated by Friedrich Gotze

Abstract In many sp ecial contexts quasiinvariance of a measure under a

oneparameter group of transformations has b een established A remarkable

classical general result of AV Skorokhod states that a measure on a

Hilb ert space is quasiinvariant in a given direction if it has a logarithmic

aj j

derivative in this direction such that e is integrable for some a

In this note we use the techniques of to extend this result to general

oneparameter families of measures and moreover we give a complete char

acterization of all functions for which the integrability

of j j implies quasiinvariance of If is convex then a necessary and

sucient condition is that log xx is not integrable at

Mathematics Sub ject Classication A C G

Overview

The pap er is divided into two parts The rst part do es not mention quasiinvariance

at all It treats only onedimensional functions and implicitly onedimensional

measures The reason is as follows A measure on R has a logarithmic derivative

if and only if has an absolutely continuous Leb esgue density f and is given by

0

f

x ae Then the integrability of jj is equivalent to the Leb esgue x

f

0

f

jf The quasiinvariance of is equivalent to the statement integrability of j

f

that f x Leb esgueae Therefore in the case of onedimensional measures a

function allows a quasiinvariance criterion as indicated in the abstract i for all

0

f

jf implies that f is absolutely continuous functions f the integrability of j

f

strictly p ositive The main result of the rst part Theorem gives necessary and

sucient reformulations of this prop erty which are easier to check The most simple

R

of these reformulations is the divergence of the integrals log xx dx where

c

is the lower nondecreasing convex envelope of Moreover we give for every with

this prop erty explicit lower estimates for the values of f on an interval I in terms

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Michael Scheutzow Heinrich v Weizsacker

R

0

f

jf dx Finally we give an j of the length of this interval and of the integral

f

I

example showing that the introduction of the lower convex hull in these results really

is necessary

The second part of the pap er then proves that the onedimensional situation is typi

cal The quasiinvariance criterion works on the real axis if and only if it works for the

transp ort of a measure under an arbitrary measurable ow or even more generally

for general one parameter families of measures which are dierentiable in the sense

of If this criterion applies then one gets even the typical CameronMartin type

formula for the RadonNikodymdensities b etween the members of such a family

cfeg In the situation of Skorokhods result mentioned in the

ax

summary we see that the exp onential functions x e a can b e replaced by

x x

exp but not by exp This shows that Skorokhods exp onential criterion

2

log x log x

is not strictly optimal but it gives the optimal p ower of log

A class of onedimensional functions

Theorem For a measurable function the fol lowing six condi

tions are equivalent

A Let f R be absolutely continuous such that

Z

f

xj f xdx j

f

and f Then f x for Lebesgueall x R

0

f

xj f x is A Let f R be absolutely continuous such that x j

f

locally Lebesgue integrable and f Then f x for al l x R

B For some a the fol lowing implication holds

X X

ai

z z z e

i i i

z

i

i i

B The implication holds for al l a

C Let be the largest nondecreasing convex function und suppose c

Then

Z

log x

dx

x

c

C Similarly lim x and for d in the range of log

x

Z

dx

log x

d

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Which Moments of a Logarithmic Derivative

In particular the conditions A B hold for if and only if they hold for If

p

is convex and nondecreasing and some power with p satises one of the

conditions then the same is true for

Proof Clearly A A

P

A B Let z and b z Dene f R by

i i

i

s z z

i

i f s exp

z

i

i i

for s z z z z i Note that e f s e

i i

0

f

and x log f x in this interval Moreover set f s for s b and

f z

i

f s f s for s Then f is absolutely continuous but not strictly p ositive

Therefore by assumption A the integral in diverges Hence

Z

z z

1 i

X X

i

f xdx e z

i

z z

i i

z z

1 i1

i i

Z

b

f

xj f xdx j

f

which proves B with a

B B Denote by B the statement that B holds with the constant a

a

P

z z Clearly B B if c b We prove B B Supp ose

i i b c a a

i

P

and let y y z for j N Then y and hence

j j j j

j

X X

ai aj a

z e y e e

i j

z y

i j

i j

From by z B C Let ht log t Dene the number z

i i

hi

x

B it follows easily that as x Thus the same holds for Since

x

is convex and increasing the function h is continuous and decreasing Therefore

P

for the pro of of it is sucient to prove that the sum z diverges

i

i

P

Supp ose on the contrary that z Cho ose y z such that y

i i

i i

i

y

i

c where

i

x

c inf

i

1

x

x



2z

i

c

i

This is p ossible by denition of this inmum c The ane function l x c x



i i i

z

i

is since it is negative on and on even the larger function x c x

 

i

z z

i i

is b ounded by Therefore from the denition of we get

c

i

l

i

z z z

i i i

We apply B with a and use the summability of the z and hence of the y to

i

i

get

X X X

i i i

y c e e e

i i

y

i

i i i

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Michael Scheutzow Heinrich v Weizsacker

i

e by construction of the z thus gives Now



i

z

i

X X X

i i

z c e z e

i

i i

z

i

i i i

which is a contradiction

C C Both and imply that is continuous nondecreasing and

unbounded at innity Therefore there is some c such that is even strictly increasing

on c and the assertion follows from lemma b elow applied to log

C A Presumably this is the most useful implication We formulate the

main part of the pro of as the separate Theorem since it involves only integrals over

nite intervals and can b e applied also to functions which do not satisfy the conditions

of the theorem In order to deduce our implication from Theorem assume and

let x x Then lim log x log x and hence using the

x

equivalence of and we get

Z

dy

log y

0

f

xjf x is lo cally integrable then Now if f is absolutely continuous and x j

f

0

f

also the function x j xjf x is lo cally integrable and hence b elow gives

f

a lower b ound for the values of f on any interval s t such that f s The case

f t follows by reection In particular f is strictly p ositive which is the assertion

of A

Finally we prove the last statement Let b e convex and nondecreasing and supp ose

p p

that satises one of the conditions If p then max and using

p

criterion B it follows that satises the same condition If p then by

p p

Jensens inequality is also convex nondecreasing and hence and

p p

Since log p log the criterion C carries over from to

In the pro of we have used the following elementary fact

Lemma Let c and let c d be a homeomorphism Then

Z Z

d x

dx dy

y x c

c d

ie both integrals converge at the same time and if they do holds

Proof The change of variables y x gives

Z Z Z

T T T

T

x x

dx dy dxx

y x x x

c

c c d

T

Since for large T the lefthand side of dominates the righthand side

T

For the converse inequality assume that the integral on the righthand side of is

nite The indenite integral on the lefthand side of is monotone in T so it has

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Which Moments of a Logarithmic Derivative

T

a nite or innite limit Therefore by the limit lim exists and it must

T

T

b e b ecause otherwise the integral on the righthand side of would b e innite

This implies

The following result gives a quantitative version of the implication C A

in Theorem

Theorem Let R be a convex even function with Let

f s t be absolutely continuous such that f s Then

min log f xf s

sxt

Z Z

t

f x

f xdx dx t s

f s f x log x

s

R R

0

y t

f x

f x dx b e nite Dene F y dx Remark Let I

1

f x log x

s

I

t s which certainly is true for y If the range of F contains the number

f s

if F y for y then can b e rewritten as

I

f t f s exp F t s

f s

This gives a lower estimate of the uctuation of the function f in terms of the integral

I and the length of the interval s t

ax

Remark In the sp ecial case x e there is an elegant more abstract pro of

of prop erty A of Theorem see prop That pro of do es not give a lower

b ound for the values of f in terms of I but on the other hand it works also in higher

dimensions whereas our metho d is strictly onedimensional

Proof Both sides of remain unchanged if f is multiplied by some p ositive

constant Therefore we may and shall for notational convenience assume f s

a

ai

inf fy s f y e g We also introduce the numbers For a i N let x

i

a a a a a

a

z x x z for i N and nally N supfn N x

n

a 1

i i i i

log ai

tg We apply Jensens inequality in the second step of the following estimates

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Michael Scheutzow Heinrich v Weizsacker

(a)

Z Z

N

a

x t

X

i

f x f x

ai

e f x dx dx

(a)

f x f x

x s

i1

i

(a)

Z

N

a

x

X

i

f x

a

ai

dx e z

i

a

(a)

f x

x

z

i1

i

i

N

a

X

a a a

ai

e z ln f x ln f x

i i i

a

z

i

i

N

a

X

a

a

ai

e z

i

a

z

i

i

N

a

X

a

a

ai

e z

i

a

z

i

i

(a) (a)

z  z

i i

y

Since is convex and the function y is increasing on

y

a

ai

e and hence the last sum can b e further estimated from b elow Moreover

(a)

z

i

by

N N

a a

X X

a

a a

ai

e z z

i i

a

z

i i

i

(a) (a) (a) (a)

z  z  z z

i i i i

N N N

a a a

X X X

a

a a

t s z z

i i

log ai

i i i

Because of

Z

N

a

b

X

a

dy

a

log ai log y

i

where

a

min log f x b lim N a lim log f x

a

N

a

sxt

a a

the pro of is complete

Example For every p there is a convex increasing function

p

which satises the conditions of Theorem but does not

This function then satises

Z Z Z

p

log x log x log x

dx p dx p dx

x x x

c c c

p

but C do es not hold for This shows that in C and in C the convex lower

p

envelope cannot b e replaced by the function itself Switching roles of and the

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Which Moments of a Logarithmic Derivative

example also shows that in the last statement of the theorem the convexity of

p

cannot b e replaced by the convexity of for p With some additional eort one

p

could mo dify the example in such a way that for no p the function satises

the conditions of Theorem

We start by setting b Construction We write q instead of

p

We shall choose recursively p oints a b a b and real numbers

k N and set

k k k

q

a for x a

k

k

q x

e for a x b

x

k k k

x for b x a

k k k k

So the function alternates b etween ane and exp onential type The constants

are chosen in such a way that at the p oints a the graph of is b ent upwards while

k

at the p oints b the two onesided derivatives agree

k

Assume that all numbers a b with i k are already chosen such that

i i i i i

gives a continuous convex increasing function on some interval b which is

k

dierentiable with the p ossible exception of the p oints a for i k In the case k

i

let a For k we then know that a and comparing logarithmic deriva

k

q q

tives of and of x resp ectively we see that x x on the interval a b

k k

q

in particular b b Since q this implies that there is a solution

k k k

k

b of the equation

k

q

x x

k k

which we choose as a Then is dened on b a by the third part of

k k k

q q

q a q a

k k

a and dene on Cho ose such that e a ie a e Let b

k k k k k

k k

a b according to the second part of The numbers and are determined

k k k k

by the equation of the left tangent of at b

k

Verification By construction

a b q b q a a

k k k k k k

ie this extension of continues to b e convex and continuous Moreover

q

3

q q

a a q b q a q

k k k k

2 2

a e b e a e e

k k

k k

The sequence a is certainly unbounded by the choice of the b By construction of

k k

q

a at this p oint the slop e of y x is bigger than Thus

k k

q q

q a b q a q a

k k k

k k

1

a

k

q 1

This implies a and hence

k

a

k 1

a

k

a

k

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Michael Scheutzow Heinrich v Weizsacker

Because of and

Z Z

a a

k +1 k +1

log x

dx log b dx

k

x x

b a

k k

q

a q log a

k k

a

a

k

k

q

for eventually all k Together with the convexity this shows that satises condition

C

P

z But implies On the other hand for z

k k

k

a

k

1

q

p

q

a z

k

k

z a

k k

P

p i p

z and therefore e So do es not have prop erty B This

k

k

z

k

concludes the discussion of the example

Logarithmic derivatives and quasi invariance

Let ME b e the linear space of nite signed measures on a measurable space E B

equipp ed with the total variation norm k k Let C b e a linear space of b ounded test

R

functions on E which is normdening for ME kk supf d C kk

g for all ME Typical examples of spaces C with this prop erty are the space of

b ounded continuous functions for a top ology for which B is the class of Baire sets ie

the eld generated by C or the space of smo oth cylindrical functions on a Hilb ert

space Let I b e a real interval and let b e a family of elements of ME We

t tI

call this map dierentiable at t I with logarithmic derivative L if for

C t t

R

every C the function s d is dierentiable at t with derivative

s

Z Z

d

d d

s t t

ds

jst

The measure is the derivative of the ME valued curve with resp ect to

t t t

the top ology ME C and is denoted by An imp ortant sp ecial class of

C

t

examples are families which are induced by a measurable ow If T T

t tR t tR

is a oneparameter group of bimeasurable bijections of E and ME is a xed

If satises the ab ove measure one considers the family of measures T

t t

t

dierentiability condition at one and then at all t we call dierentiable along T

with logarithmic derivative In this case the logarithmic derivative for general

t is given by

x T x

t t

This extends the concept of the dierentiability of a measure on a linear space in

a certain direction which was the main sub ject of and the relevant parts of

The more general asp ects have b een studied starting with in and for a

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Which Moments of a Logarithmic Derivative

comparison with concepts of the GrossMalliavin calculus see eg

We need two results from a Supp ose that exists for all t I and that

t

Z Z

dt k k dt k k

t

t t

I I

Then there are a probability measure on B and B B I measurable functions g g on

0

g tx

E I such that dx g t x dx dx g t x dx and thus x

t t

t

g tx

ae for Leb esgue almost all t I and nally

Z

b

g s xds f or al l x E and a b I g b x g a x

a

b If moreover the p ointwise integrability condition

Z

b

j xj ds j j j j ae

s a b

a

holds then all measures a t b are equivalent and we have the abstract

t

CameronMartin formula

Z

b

d

b

x exp x ds

s

d

a

a

The condition clearly is necessary for to make sense But how can one

verify it The interaction of the RadonNikodym derivatives for varying t may

t

b e complicated Therefore it seems desirable to have sucient conditions for the

equivalence of the in terms of the onedimensional laws of the with resp ect to

t t

the measures The following continuation of the main result of the rst section

t

provides an answer of this type

Theorem A function satises the conditions A C of

Theorem if and only if the fol lowing holds For every dierentiable and k k

C

bounded family I R of signed measures on a measurable space and a b I

t tI

with

Z

b

dt k j jk

t

t

a

the measures a t b are equivalent to each other Moreover for such functions

t

the condition implies the abstract CameronMartin formula formula

Proof Supp ose that has the indicated prop erty We want to show that

condition A of Theorem is fullled Let f b e an absolutely continuous nonnegative

0

f

xj f x is lo cally integrable and such function on the real axis for which x j

f

that f do es not vanish everywhere We have to show that f is strictly p ositive

Otherwise there are two p oints a b with f a and f b Without loss of

generality a b Let dx f x tdx In order to apply our condition we have

t

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Michael Scheutzow Heinrich v Weizsacker

to make sure that these measures are nite For this redene f on b by f x

and on a by f x f a expx a Then

Z Z

a a

f

xjf x dx f a expx a dx j

f

R

0

f

and similarly j xjf x dx The mo died f still satises and

f

b

it is certainly Leb esgue integrable Thus we have the ow situation mentioned ab ove

with T x x t The family is dierentiable even for the top ology induced by the

t t

0

f

x t Then the lo cal integrability assumption total variation norm with x

t

f

R R

and the two tail estimates imply j xj dx j xj dx for all

t t

t Therefore the condition is satised By our assumption on this implies that

the measures are all equivalent ie the function f cannot vanish on a halfline as

t

our f do es This contradiction shows that f must b e strictly p ositive Hence has

prop erty A

Supp ose conversely that is a function of the type considered in Theorem Let

b e a dierentiable and k kb ounded family I R of signed measures

t C t tI

on a measurable space and let a b I with b e given First we claim that

holds In fact from condition C in Theorem we nd p ositive constants u v such

that v y y for all y u Then

Z Z Z

u dj j j j dj j j xj d v k k

t t t t t t

t

E j ju E

t

uk k v k j jk

t t

t

Since the measures are k kb ounded implies Therefore we can choose g g

and with the prop erties listed after Then can b e rewritten as

Z Z

b

g t x

jg t x dx dt j

g t x

a E

0 R

b

g

t

j By Fubini there is a nullset N such that xjg x dt for every

t

g

a

t

x E n N Then extending t g t x outside of the interval a b by exp onential

tails or zero as in the rst part of this pro of we can apply condition A in Theorem

and conclude that for each x E n N either g t x for all t a b or g t x

for all t a b This implies that the measures t a b are all equivalent

t

Moreover the function g x is continuous by and therefore it is b ounded

away from by some constant a b x on the interval a b for and almost

a b

0

g tx

show that and hence all x E Then and the representation x

t

g tx

also hold

In particular we get the following version of Skorokhods theorem for the function

y

given by y exp for y

j log y j

Corollary Let be a probability measure on a measurable space E and let T

T be a measurable ow on E Suppose is dierentiable along T with

t tR C

logarithmic derivative If satises the fol lowing integrability condition

Z

jxj

dx exp

j logjxjj

E

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Which Moments of a Logarithmic Derivative

then is quasiinvariant under the ow T and the corresponding RadonNikodym

derivatives are given by But even for translation families on the real axis the

quasiinvariance is not implied by the weaker condition

Z

jxj

dx exp

log jxj

E

y

Proof We consider the function y exp for y Then it is easily

j log y j

veried that is convex and increasing for suciently large y and thus it satises

the criterion C Because of we have

Z Z

jj T d jj d k j jk

t t

t t

for all t and hence our integrability assumption implies

y

On the other hand y exp for y denes a function which do es

2

log y

not satisfy the condition C The function f used in the pro of of A B in

Theorem then satises for this function but f has compact supp ort Therefore

0

f

of the measure MR whose density is f the logarithmic derivative

f

satises the weakened integrability condition of our Corollary but this measure is not

quasiinvariant under the ow of translations

Acknowledgement This work was motivated by stimulating discussions of the

second author in the fall of at the Mathematical Science Research Institute

Berkeley in particular with D Bell and O Zeitouni who raised questions which are

answered in Theorems und

References

S Alb everio and M Rockner Classical Dirichlet forms on Topological Vector

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V I Averbukh O G Smolyanov and S V Fomin Generalized functions and

dierential equations in linear spaces I Dierentiable measures Trudi of Moscow

Math Soc

D R Bell A quasiinvariance theorem for measures in Banach spaces Trans

Am Math Soc

V I Bogachev and E MayerWolf Absolutely Continuous Flows Generated by

Sob olev Class Vector Fields in Finite and Innite Dimensions Universitat Biele

feld SFB preprint

Daletski Yu L and Sohadze G Absolute Continuity of Smo oth Measures Func

tional Analysis and Applications N

A V Skorokhod Integration in Hilbert Space Springer Heidelb ergNew York

Documenta Mathematica

Michael Scheutzow Heinrich v Weizsacker

O G Smolyanov and H v Weizsacker Dierentiable families of measures Journal

of Functional Analysis

O G Smolyanov and H v Weizsacker Dierentiable measures and a Malliavin

Stro o ck theorem submitted

Michael Scheutzow Heinrich v Weizsacker

Fachbereich Mathematik Fachbereich Mathematik

der Technischen Universitat der Universitat

Strae des Juni D Kaiserslautern

D Berlin Germany

Germany weizsaeckermathematikuniklde

msmathtub erlinde

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