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On logarithmic Sobolev inequalities�

Csisz�ar�Kullback inequalities� and the rate of

convergence to equilibrium for Fokker�Planck

type equations

June �� ����

Anton Arnold�� Peter Markowich�� Giuseppe Toscani��Andreas Unterreiter�

�Fachbereich Mathematik� TU�Berlin� MA ���� D����� Berlin� Germany

�Institut fur Analysis und Numerik� Universitat Linz� A��� Linz� Austria

�Dipartimento di Matematica� Universit�a di Pavia� I���� Pavia� Italy

�Fachbereich Mathematik� Universitat Kaiserslautern� D���� � Kaiserslautern�Germany

arnold�math�tu�berlin�de� markowich�numa�uni�linz�ac�at�toscani�dimat�unipv�it� unterreiter�mathematik�uni�kl�de

Acknowledgements

This research was partially supported by the grants ERBFMRXCT��� � �TMR�Network� from the EU� the bilateral DAAD�Vigoni Program� the DFG underGrant�No� MA ��������� and the NSF �DMS�� � ��� The �rst author ack�nowledges fruitful discussions with L� Gross and D� Stroock� the second authorwith D� Bakry� and the second and third authors interactions with C� Villani�

AMS ���� Subject Classi�cation� � K �� � B�� ��D�� ��D�� �J�

Abstract

It is well known that the analysis of the large�time asymptotics of

Fokker�Planck type equations by the entropy method is closely related to

proving the validity of logarithmic Sobolev inequalities� Here we highlight

this connection from an applied PDE point of view�

In our uni�ed presentation of the theory we present new results to thefollowing topics� an elementary derivation of Bakry�Emery type condi�

tions� results concerning perturbations of invariant measures with gene�

ral admissible entropies� sharpness of convex Sobolev inequalities� explicit

construction of optimal Csisz�ar�Kullback inequalities� applications to non�

symmetric linear �e�g� kinetic� and certain non�linear Fokker�Planck type

equations �Desai�Zwanzig model� drift�diusion�Poisson model�

Contents

� Introduction �

� Decay of the relative Entropy as t�� �

��� Spectral gap of the symmetrized equation � � � � � � � � � � � � � �

��� Admissible relative entropies and their generatingfunctions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Exponential decay of the entropy production and the relative entropy ��

��� Convex Sobolev inequalities in entropy version � � � � � � � � � � � ��

�� Non�symmetric Fokker�Planck equations � � � � � � � � � � � � � � �

� Sobolev Inequalities ��

��� Three versions of convex Sobolev inequalities � � � � � � � � � � � � ��

��� Perturbation lemmata for the potential A�x� � � � � � � � � � � � � �

��� Poincar�e�type inequalities � � � � � � � � � � � � � � � � � � � � � � ��

��� Sharpness results � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� A generalized Csisz�ar�Kullback Inequality ��

��� Optimality � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

��� An extension of the inequality � � � � � � � � � � � � � � � � � � � � �

��� Comparison with the classical Csisz�ar�Kullback inequality � � � � �

��� An example � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Asymptotic behavior of U as e��u�� � � � � � � � � � � � � � � �

��� Passing from weak to strong convergence in L��d�� � � � � � � � � �

��� The optimal inequality for the logarithmic entropy � � � � � � � � � �

��� Entropy�type estimates on kf � gkL��d�� � � � � � � � � � � � � � � �

� Nonlinear Model Problems �

�� Desai�Zwanzig type models � � � � � � � � � � � � � � � � � � � � � � ��

�� The drift�di�usion�Poisson model � � � � � � � � � � � � � � � � � � ��

Appendix Proofs of Section � ��

� Introduction

One of the fundamental problems in kinetic theory is the analysis of the timedecay �rate� of solutions of kinetic models towards their equilibrium states� Themaybe most often applied methodology in time asymptotics is the entropy me�thod� where the convergence towards equilibrium is concluded using the timemonotonicity of the physical entropy of the system� A classical example for thisapproach is provided by the spatially homogeneous Boltzmann equation� whereconvergence to the Maxwellian equilibrium state has been proven in this way �cf��e�g�� �CeIlPu���� Theorem ����� and the references therein��

To illustrate the entropy approach we shall now present two simple �even expli�citly solvable� kinetic model problems outlining the ideas which we will developin this paper�

At �rst we consider the Bhatnagar�Gross�Krook �BGK� model of gas dynamics�which is a �simpli�ed� version of the Boltzmann equation carrying to some extentthe same amount of physical information �BhGrKr ��� The IVP for the spacehomogeneous version reads�

�f

�t� J�f� �� �

�Mf � f� � v � IRn� t � � ����a�

f�v� t � � � fI�v�� v � IRn ����b�

�of course� only the case n � � is of physical interest�� with the normalizationRIRnfIdv � �� Here M

f �Mf �v� denotes the Maxwellian distribution function

Mf �v� � m ������n�� exp��jv � uj

��

������

with mass

m �

ZIRn

fdv�

mean velocity

u �

ZIRn

vfdv�m

and temperature

� �

ZIRn

�v � u��f�nm�

The relaxation rate � is assumed to be a positive constant� It is then an easyexercise to show that m� u and � are left invariant under the temporal evolutionof ������ such that these quantities can be computed from the initial data fI � andMf�t� �MfI follows�

The exponential L��IRnv ��convergence of f��� t� to its equilibrium state MfI with

rate � can be easily veri�ed by explicitly solving ������

For carrying out the entropy approach we consider the physical entropy �Boltz�mann�s H�functional�

H�f� �

ZIRn

f ln fdv �����

and the entropy production

I�f� �

ZIRn

ln fJ�f�dv� �����

with the easily veri�able relation

d

dtH�f�t�� � I�f�t��� ��� �

Since lnMf is quadratic in v and due to the conservation of mass� mean velocityand temperature� we calculate

I�f�t�� � �

ZIRn

ln f�t��Mf�t� � f�t�� dv

� ��ZIRn

�ln f�t�� lnMf�t�

� �f�t��Mf�t�

�dv

� ��ZIRn

ln

�f�t�

Mf�t�

�f�t�dv � �

ZIRn

ln

�Mf�t�

f�t�

�Mf�t�dv�

Since f�t� and Mf�t� have the same mass� Jensen�s inequality implies that bothterms are nonpositive� Thus we obtain the stronger version of Boltzmann�s H�Theorem

�I�f�t�� � �e�f�t�jMf�t�� �����

where the so called relative �to the Maxwellian� entropy is de�ned by

e�f jMf� ��

ZIRn

f ln

�f

Mf

�dv� �����

Note that e�f jMf� � i� f � Mf �Gibb�s Lemma�� Again� since lnMf isquadratic in v and since f�t� and Mf�t� have the same mass� velocity and tem�perature� we have

e�f�t�jMf�t�� � H�f�t���H�Mf�t���

and since Mf�t� �MfI we conclude from ��� �� �����

d

dte�f�t�jMf�t�

��d

dtH�f�t�� � I�f�t�� � ��e �f�t�jMf�t�

��

Exponential convergence to zero of the relative entropy with rate � follows im�mediately �for initial data which have �nite relative entropy�

e�f�t�jMfI � � e��te�fI jMfI �� t � � �����

The Csisz�ar�Kullback inequality �Csi�����Kul ��

kf �Mfk�L��IRn� � �e�f jMf � �����

then gives L��IRn��convergence to equilibrium �at the suboptimal rate ����

Summing up� we used the lower bound ����� of the �negative� entropy productionin terms of the relative entropy to explicitly control the convergence of the �rela�tive and absolute� entropies and to control the strong convergence of the solutionto its equilibrium state� The second model problem� which is a two�velocity ra�diative transfer model� demonstrates that the approach of the �rst example maynot be general enough� We consider the ODE in IR��

d�

dt�

� �� �� ��

��� t � ����a�

��� �

�uIvI

�����b�

where ��t� �

�u�t�v�t�

�and � Since we consider ����� as a kinetic model�

we assume � for physical reasons only � uI� vI � which implies u�t�� v�t� � �Obviously� u�t� and v�t� tend exponentially with rate � to their equilibriumstate w� � �uI vI���� To apply the entropy approach let � � ��s� be anysmooth strictly convex function� de�ned on IR�� with ���� � � We introducethe relative entropy generated by ��

e���j��� �����u

w�� ��

v

w��

�w� ������

where we denoted the steady state

�� ��w�w�

��

Di�erentiating ������ with respect to t and using ����� gives the entropy equation

d

dte����t�j��� � I����t�j��� ������

with the production term

I���j��� � ��u� v�����u

w��� ��� v

w��

�� ������

Since � is strictly convex we have I���j��� � with equality i� � � ���Clearly� now we cannot bound the entropy production ������ from below directly

by a multiple of the relative entropy ������� Therefore we consider the timeevolution of the entropy production� Di�erentiating ������ gives

d

dtI����t�j��� � R����t�j��� ������

with the entropy production rate

R����t�j��� � ��I����t�j��� � �u�t�� v�t���

w�

�����u�t�

w�� ����

v�t�

w��

��

���� �

We can now bound the entropy production rate from below by a multiple of theentropy production�

R����t�j��� � ��I����t�j���

and conclude exponential convergence with rate � of the entropy productionfrom ������

jI���t�j���j � jI��I j���je���t� ������

Inserting I����t�j��� from ���� � into ������ and using ������ gives after integra�tion from s � t to s �� �using limt�� e���t�j��� � limt�� I����t�j��� � �

e����t�j��� Z �

t

�u�s�� v�s����w�

�����u�s�

w�� ����

v�s�

w��

�ds � �I���t�j���

��

������

First of all� ������ gives the exponential decay with rate � of the relative entropye�� and hence again the suboptimal decay rate of ��t� towards ��� Moreover������� furnishes an inequality involving the strictly convex function �� i�e� weobtain the Sobolev�type inequality�

��uIw�

� ��vIw�

�w� � �

��uI � vI�

����uIw�

�� ��� vIw�

�������

for all uI� vI � by setting t � in ������� Also� ������ implies that equality in������ holds i� u�t� � v�t� for all t � which is equivalent to uI � vI � w��Clearly� the inequality ������ can easily be obtained in a direct way �cp� Example��� of �Gro����� however� the presented approach allows generalization to muchmore complex kinetic problems �as will be the subject of the next sections��Particularly� the second example clearly shows that in some cases the entropyequality itself is not su!cient to derive exponential decay of the relative entropy�and equivalently� Sobolev�type inequalities�� It may be necessary to consider

the evolution of the entropy production and to �nd a lower bound of the entropyproduction rate in terms of the entropy production�

This leads directly to the main subject of this paper� In the following sectionswe shall consider the IVP for Fokker�Planck type equations �cf� ������ and applythe methodology presented above�

More speci�cally� Section � will be concerned with the decay of the relativeentropy as t��� There we shall obtain conditions on the entropy� the di�u�sion matrix and on the velocity �eld which allows for exponential convergenceunder weak conditions on the initial data �bounded relative entropy�� Section �then is concerned with the derivation of various forms of Sobolev�type inequalitiesand with the analysis of conditions which guarantee that the inequalities �satu�rate� nontrivially� In Section � we generalize the Csisz�ar�Kullback inequality torather general convex entropies� and in Section we apply the developed theoryto nonlinear Fokker�Planck models�

We conclude this introduction with a brief �and incomplete� discussion of therelevant literature and of those issues where our approach di�ers from alreadyexisting ones� Most importantly� our approach is to some extent based on thework by D� Bakry and M� Emery �cf� e�g�� �BaEm���� �Bak���� �Bak����� whichprovides a very general framework for hypercontractive semigroups and generali�zed Sobolev�type inequalities using the so�called iterated gradient �"�� formalism�BaEm���� In fact� some of our results are concretizations of the Bakry�Emery cri�terion �cf� the references cited above� to Fokker�Planck type equations� However�the Bakry�Emery approach does not allow an explicit control of the remainderterm in the Sobolev inequalities� Let us explain this point in more detail� Care�ful reading of the radiative transfer�type example presented above shows that theanalysis of the time decay of the entropy production gives at the same time the#sharp$ decay of the relative entropy as well as the remainder in ������� Theknowledge of this remainder allows to identify in ������ the �unique� state sa�turating the Sobolev�type inequality� Hence� the entropy approach gives at thesame time a proof of a #convex$ inequality and all cases of equality�

As far as the classical Gross logarithmic Sobolev inequality is concerned �Gro� ��the identi�cation of the cases of equality is due to Carlen �Car���� His approachis based on a slightly di�erent method� which relies on information�type ine�qualities� and ultimately requires a deep investigation of properties of Gaussianfunctions� In the same paper� the connection between Gross� logarithmic Sobolevinequality and the linear Fokker�Planck equation� through the Fisher measure ofinformation and the Blachman�Stam inequality �Bla� �� �Sta �� has been fruit�fully developed� The entropy�entropy production approach for the same Fokker�Planck equation has been recently addressed in �Tos��b�� �Tos��a�� There� thecases of equality follow easily from the identi�cation of the remainder� Physicallyspeaking� the entropy approach put in evidence that the remainder in this type

of inequalities depends on the complete dynamics in time of the solution of the�mass�conserving� linear problem that generates the inequality itself� In otherwords� many Sobolev type inequalities can be interpreted as an estimate for therelative entropy of the initial state with respect to the steady state of a linearmass conserving system�

An excellent background reference for logarithmic Sobolev inequalities is theoverview paper �Gro���� General linear homogeneous radiative transfer equa�tions generating certain inequalities involving convex functions are analyzed in�GaMaUn���� A calculation of the logarithmic Sobolev constant for non�symmetricODEs in IR� �generalizing our second example above� is presented in the appen�dix of �DiSC���� Entropy production arguments for the decay of the solution ofthe heat equation in IRn towards the fundamental solution were used in �Tos��a��A further application in kinetic theory is to be found in the asymptotic analysisof the spatially homogeneous Boltzmann equation� which in the so called �gra�zing collisions limit� leads to the Landau�Fokker�Planck equation �Vil���� �Vil�����Tos��b�� There it is hoped that the decay results of the relative entropy mightgive an indication of the rate of decay towards equilibrium of the spatially homo�geneous Boltzmann equation�

� Decay of the relative Entropy as t��

We now consider the IVP for the Fokker�Planck type equation

�t � div�D�r� �rA��� x � IRn� t � ����a�

��t � � � �I � L��IRn�� ����b�

with A � A�x� su!ciently regular �i�e� A � W ���loc �IR

n� and e�A � L��IRn��We assume that the symmetric di�usion matrix D � D�x� � �dij�x�� is locallyuniformly positive de�nite on IRn and dij � W ���

loc �IRn�� i� j � �� � � � � n� Obviously

we have the conservation propertyZIRn

��x� t�dx �

ZIRn

�I�x�dx� �����

In a kinetic context� the independent variable x in ����� stands for the velocity�

In this Section we assume �without restriction of generality�ZIRn

�I�x�dx � ��

One easily sees that ����a� has the steady state

���x� � e�A�x� � L���IRn�� �����

assuming �w�r�o�g�� A to be normalized asRIRne�A�x�dx � �� We remark that �by

a simple minimum principle� �I�x� � implies ��x� t� � for all x � IRn� t � �In this Section we shall investigate the convergence of ��t� towards the steadystate �� in various norms and in relative entropy� In particular we are concernedwith equations ����a� that exhibit an exponential decay to the steady state ������

��� Spectral gap of the symmetrized equation

We transform equation ����� to symmetric form �on L��IRn��� Therefore we set

z �� ��p���

which satis�es the IVP

zt � div�Drz�� V �x�z� x � IRn� t �z�t � � � zI �� �I�

p�� on IRn� �����

Here� V �x� denotes the potential

V �x� � ����Tr�D

��A

�x��� �

��rA��DrA �divD� � rA�� x � IRn� t �

��� �

��A�x�

is the Hessian of A�x� and the superscript #�$ denotes transposition� Nowde�ne the Hamiltonian

Hz � �div�Drz� V z �����

on the domain

DQ �� fz � L��IRn�jQ�z� z� �g

of the quadratic form Q�z�� z�� �� �Hz�� z��L��IRn� given by

Q�z�� z�� ��

ZIRn

r��z�p��

�D�x�r

�z�p��

����dx�

�obtained from ����� by a simple calculation�� For the following we shall assumethat the di�usion matrix D�x� and the potential A�x� are such that H generatesa C��semigroup on L��IRn� �for various su!cient conditions see �ReSi����� Notethat H satis�es

Hz �p��N

�zp��

�� z � DQ�

where N is the Dirichlet form�type operator on L��IRn� d��� de�ned by

�Nu� v�L��IRn�d��� ��

ZIRn

r�uDrv ���dx�

�see �Gro����� For future reference we also remark that

�ZIRn

����L�� dx � Q

���p�����p��

��

��p�����p��

� DQ� �����

Here L denotes the appropriate extension of the Fokker�Planck type operator

L� �� div �D �r� �rA�� �Obviously the spectrum ��H� is contained in ����� Since Q�z� � i� z �const

p�� we conclude that the ground state z� � exp��A��� of H is non�

degenerate and that �� is the unique normalized steady state of ����a��

The solution of ����� can be written as

z�t� �p��

Z�����

e��td�P��Ip���� �����

where P� is the projection valued spectral measure of H� Of course� we assumehere �I�

p�� � L��IRn�� From this spectral representation we immediately con�

clude the convergence of z�t� top�� in L��IRn� as t��� Let us now consider

a HamiltonianH with a positive spectral gap � �i�e� distance of ��H�nfg fromthe eigenvalue �� For the case D�x� � I� the identity matrix� a simple su!cientcondition for a positive spectral gap is given by V � L�loc�IRn% IR�� V �x� boundedbelow and V �x��� for jxj� � �see e�g� �ReSi��a� Th� XIII������ In this casewe obtain exponential convergence�

kz�t��p��kL��IRn� � kzI �p��kL��IRn�e�t�� � �����

which implies exponential convergence of ��t� to �� in L��IRn��ZIRnj��t�� ��jdx �

ZIRn

p��j��t�� ��jp

��dx

� k��k��

L��IRn�kz�t��p��kL��IRn� � O�e

�t���� �����

This very simple approach of course only holds under the restrictive assumption�I�

p�� � L��IRn�� In order to weaken this restriction and to better illustrate the

convergence of ��t� towards �� we now investigate the decay in relative entropye���t�j��� de�ned by ������ In fact� we shall not only consider this logarithmic#physical entropy$� but a wider class of relative entropies that essentially lie�between� this physical entropy and the quadratic functional k��t�k�

L��IRn����� �dx���

��

For future reference we also consider a second symmetrization of ����� �on L��IRn� d����which is more commonly used in the probabilistic literature on this subject� Weset

� �� �����

which satis�es the IVP

�t � ���� div�D��r��� x � IRn� t �

�I �� �I��� � L��IRn� d���� ������

In terms of this symmetrized problem we shall extend the allowed initial data �Ifrom L��IRn� ��� to the Orlicz space L� logL�IR

n� ����

��� Admissible relative entropies and their generating

functions

De nition ���� Let J be either IR or IR�� Let � � C�J� C��J� �where Jdenotes the closure of J� satisfy the conditions

���� � � �����a�

��� � � ��� � on J� �����b�

������� � �

�����IV on J� �����c�

Let �� � L��IRn�� �� � L���IRn� withR��dx �

R��dx � � and ����� � J ���dx��

a�e� Then

e����j��� ��ZIRn������� ���dx� ������

is called an admissible relative entropy �of �� with respect to ��� with generatingfunction ��

Note that

e����j��� � ������

follows from Jensen�s inequality�

��

Remark ���� If � satis�es the conditions �����a������c� so does its �norma�

lization� e���� � ���� � ������� � �� and they both generate the same relativeentropy� e

e����j��� � e����j���� In the sequel we will therefore assume that thegenerator � of e� be normalized as

����� � � �����d�

This implies that the relative entropies e� and their generators � are bijectivelyrelated� Due to the convexity of � we have

� � on J� ���� �

We now list some typical examples of admissible relative entropies on J � IR��The physical relative entropy ����� is generated by ���� � � ln� � � � ratherthan by ���� � � ln�� It is a special case of

���� � ��� �� ln� �

� �� ��� � ��� � � � � � �����a�

For � p �

�p��� � ���� ��p � �� ��p � p�� ��p���� � ��� � � � � � � �����b�

and for p � �

���� � ��� � ���� � � �����c�

generate admissible relative entropies� In the last example we clearly have e���j��� ���k��k�L��IRn����� �dx��

� ���In the above de�nition we excluded linear entropy functionals as they would bezero due to the assumed normalizations of �� and ���

In the following Lemmata we derive important properties of admissible entropies�

Lemma ���� a� Admissible entropies are generated by strictly convex functions�� b� For J � IR all admissible entropies are generated by ���� c��

Proof� a� Let g��� �� ������

� � � J with g��� � ����� Then� condition�����c� is equivalent to

g����� � ������

whenever ������ � Since ������ excludes �positive� poles of g on J � we conclude��� on J �

b� g�� � and g on IR implies g��� � const � and the assertion follows�

��

Hence� our class of generating functions � coincides with those considered in�BaEm��� �up to the normalizations �����a�� �����d���

For the following we shall assume J � IR� and �I � � Except being reasonablefrom a kinetic viewpoint� this case allows for a richer mathematical structuresince only quadratic admissible entropies exist for J � IR�

The above examples �����a�� �����c� of admissible entropies include the two limi�ting cases for the asymptotic behavior �as � ��� of the generating function ��An admissible relative entropy e����j��� can be bounded below by a logarithmicsubentropy e� and bounded above by a quadratic superentropy e�

Lemma ���� Let � generate an admissible relative entropy with J � IR�� Thenthere exists a logarithmic�type function � ���� a� and a quadratic function ����� c� such that

���� � ���� � ����� � � J� �����a�

and hence

� e����j��� � e����j��� � e���j���� �����b�

� and � both satisfy ������ and thus generate respectively an admissible sub�and superentropy for e��

Proof� Since J � IR�� the function g from the proof of Lemma ��� satis�es

g � g� � � g�� � on J� ������

Now denote the derivatives of the given function � by

���� � � ����� � � ������ �� �� � ������� �� �� � � �����

From ������ we readily get the estimate

����� � � ����� � � �

�� g��� � �� �� � �

with � �� ������� � � � �� ��� ������� � � Integrating the corresponding

estimate for ��� � �g�

��� ���� � ������ ������� � ���� � �

������

we obtain with ����� the upper bound for ��

���� ������ ln� � � ��� � �

����� � ���� � �

�� ���� � ��� �� ����� � � ������

��

To derive the lower bound of � one integrates ������ twice to show ���� � �����For � the function ���� is given by �����a� with � � �

�� � �

��

If � � we set

���� ������ � ���� ������

The well�known Csisz�ar�Kullback inequality ��Csi���� �Kul ��� shows that thelogarithmic relative entropy ����� is a �measure� for the distance between twonormalized L���IR

n��functions ��� �� withRIRn��dx �

RIRn��dx � ��

�k�� � ��k�L��IRn� � e���j���� ������

In a simple calculation using the subentropy e� and Lemma ��� this inequalitycan be extended to any admissible entropy e�� For � � and � hence givenby �����a� we introduce the normalized function e� �� ���� ��

�� � � and estimate

using �������

�k�� � ��k�L��IRn� �

�������

ke�� ��k�L��IRn� �������e�e�j��� � �

��e����j���� ���� �

With �����b� this gives

�k�� � ��k�L��IRn� �

��e����j���� ������

with the notation �� � ������ � ������� If � � � ������ is easily derived with

the estimate ����� and with ������� A detailed analysis of Csisz�ar�Kullbackinequalities for a larger class of generating functions � will be the topic of x�� Inparticular we shall derive a sharper variant of �������

In our subsequent analysis we shall need the continuity of the relative entropy�

Lemma ���� Let the sequence �j � � �as j ��� in L���IRn� ���� �dx�� with the

normalizationR�jdx �

R�dx �

R��dx � �� Then for all admissible entropies

� with J � IR��

e���jj���� e���j��� as j��� ������

Proof� Since � � C�� �� there exists for any � a positive � � ���� such that

j������ �����j � for � ���� � ����� ������

Integrating ������ we readily obtain the estimate�

�C���� such that j�����j � C������ �� for � � �� ������

We shall �rst derive estimates on j��a�� ��b�j% a� b � J � for three di�erent casesof a and b� For a� b � � we use the mean value theorem and the monotonicity of�� to estimate�

j��a�� ��b�j � ja� bj�j���a�j j���b�j� � ja� bjC������ a b�� �����

For the next case assume a � �� b � �or vice versa�� With ����� this yields

j��a�� ��b�j � j��a�� ����j j����� ��b�j� �a� ��C������ a �� �� � b�C����

� ja� bj �C������ a b �� C����� � ������

with C���� �� supc��� �j�� ����c�j

�c ��Finally� for a� b � we have j��a�� ��b�j � from �������Using these three estimates we can now control the di�erence of relative entropies�For arbitrarily small � we estimate�

je���jj���� e���j���j�

Zf �j��

� or ���

� g

�� �j���� �

��

��

� ��dx ������

Zf �j��

� ���

� g

�� �j���� �

��

��

� ��dx� �C������ �� C�����

ZIRn

j�j � �jdx ������

C����

ZIRn

j�j � �jp��

�j �p��dx �

ZIRn

��dx

� �C������ �� C����� k�j � �kL��IRn�

C����k�j � �kL��IRn����� �dx��k�j �kL��IRn����� �dx�� �� ������

As j�� this last term converges to �� since L��IRn� ���� �dx���convergence impliesL��IRn��convergence� This �nishes the proof�

For future reference we state the following elementary result for generators ofadmissible relative entropies�

��

Lemma ��� The generator � of every admissible relative entropy e� satis�es�

a�

����

� ��

��� �

�� ����� � �����

� ��

��� �

�� � � ���� �

with the notation �� � ������

b�

���� � �����

��

��

��

��

��

��� �

��� � ��� � � �� � ������

���� � ������

�� ��

��

��� �

��� � ��� �� � � � ������

Proof� a� For the right inequality of ���� � we have to show that G���� ���� ���� � �� � ��� � � But this follows from G���� � G

����� � � G

������ �

������ � � � �see �������For the left inequality of ���� � we have to show that G���� �� � ���� � ������ � � Like before we have G���� � G

����� � and it remains to show

G������ � ���� � ����� � � � � ������

Since � generates an admissible entropy we have ������� � � �IV ��� � �see������� �����c��� Thus there exists a unique �� � ���� such that ������� on�� ��� and �

������ � on �������In the case � � ������ we have G������ � ������� �In the case � � �� ��� we rewrite �����c� as

� � �����

����

��and integrate over the interval ��� ��� � ���

� � � � � �������������

�������������

� � �������������

Letting tend �� we obtain G��� � ���� � ����� � � and this �nishes the proofof ���� ��

b� Applying the Gronwall lemma to the right inequality of ���� � �with �xed ���gives

���� � �������

��

��

��

��

��� �

��� � ���� �

�� � � �� �

��

and ������ follows from �� ��

For the proof of ������ we use the left inequality of ���� � and estimate�

����� � �����

��

��� �

�� �����

��

��

�� �

��

�� � � ������

Applying the Gronwall lemma to ������ gives

���� � ����� ��� ���� ln

�� �� �

���� � � ���

and ������ follows with the estimate lnx � x� ��

��� Exponential decay of the entropy production and the

relative entropy

With our notion of superentropies we can now show the convergence of ��t� to�� in relative entropy�

Lemma ���� Let e� be an admissible relative entropy and assumezI � �I�

p�� � L��IRn�� Then e����t�j���� as t���

Proof� With the notation �� � ������ we estimate using �����b�� �������

� e����t�j��� � e���t�j��� �ZIRn

���t���

����dx� � ��kz�t��p��k�L��IRn��

�����

and the assertion follows from ������

If the Hamiltonian H in ����� has a spectral gap � � then the exponentialdecay from ����� of course carries over to the relative entropy� In the above lemma�the assumption �I�

p�� � L��IRn� is unnaturally restrictive� The subsequent

analysis aims at considering initial data which only have �nite relative entropyand at proving explicit decay bounds for the relative entropy in terms of theinitial relative entropy� The �price� for this extension will be a condition on theevolution problem ����a� �see �A�� below� that is stronger than assuming H tohave a spectral gap�

We now proceed similarly to �Tos��b� and to example � of the introduction�Consider the entropy production

I����t�j��� �� ddte����t�j��� ������

��

and the entropy production rate

R����t�j��� �� ddtI����t�j���� ������

To facilitate the computations we rewrite ����a� in the following form�

�t � div�Du��� ������

with the notation u � r� ����� Di�erentiating the relative entropy e����t�j���

gives

I����t�j��� �ZIRn�� ���

��t dx� ������

By using ������ we obtain after an integration by parts

I����t�j��� � �ZIRn

��� ���

�u�Du�� dx � � ���� �

due to the positivity of D� Using ������ we compute �������

R����t�j��� � �ZIRn

���� ���

�div�Du���u

�Du dx

� �

ZIRn

��� ���

�u�Dut��dx� ������

Clearly� the computations which lead to ���� � and ������ are formal� However�they can easily be justi�ed for initial data �I � L��IRn% ���� �dx�� and for entropygenerators without singularity at � � by taking into account the semigroupproperty of the HamiltonianH and the partial integration formula ������ Generaladmissible entropies can easily be dealt with by a local cut�o� at � � �

We now return to proving the exponential decay of e����t�j��� under additionalassumptions on A and D� At �rst we shall derive an exponential decay rate forthe entropy production I� by using the special form of the entropy productionrate �������

At �rst we consider the case of a scalar di�usion� i�e� D�x� � ID�x��

Lemma ���� Let the initial condition �I � L��IRn% ���� �dx�� satisfy jI���I j���j � for the admissible entropy e�� Assume that the scalar coe�cients A�x� andD�x� of ����a� satisfy the condition

�A�� �� such that��� n�

� �DrD �rD �

��&D �rD � rA�I

D��A

�x� rA�rD rD �rA

�� �

�D

�x�� �I

��

�in the sense of positive de�nite matrices� x � IRn� Then the entropy productionconverges to � exponentially�

jI����t�j���j � e����tjI���I j���j� t � ������

Proof� After an integration by parts �which can be justi�ed as mentioned above�the �rst term of the entropy production rate ������ reads

R� �

ZIRn

�IV�eA��D�juj�e�Adx

ZIRn

�����eA��D�e�Au��u

�xudx

ZIRn

�����eA��e�ADjuj�u � rDdx�

We set

ut � r�eAdiv�De�Au��in the second term of ������� which becomes

R� � ��ZIRn����eA��De�A�u � r�Ddivu� u�r� �rD �rAD�u u��u

�x�rD �rAD��dx�

We di�erentiate u � r�Ddivu� � �u � rD�divu Du � r�divu�� and we expressDu � r�divu� according to

�&�Djuj�� � �

�juj�&D �rD��u

�xu D

Xi�j

��ui�xj

��

Du � r�divu��

We rewrite R� as R� � S� S� with

S� � �ZIRn����eA��div�Dr�Djuj��e�A�dx

ZIRn

�����eA���D�u � r�juj�� DrD � ujuj�� e�Adx�

S� � �

ZIRn

����eA��De�A�u�r� �rAD �rD�u

�&Djuj� � �

�juj�rD � rA �

D

�� n��u � rD��

dx

ZIRn

����eA��e�A�n� ���u � rD�� �D�u � rD�divu

�DrD��u�xu D�

Xi�j

��ui�xj

��

�juj�jrDj�

dx

� T� T��

The second integral in S� can be written as

T� � �

ZIRn

����eA��e�AXi�j

�D�ui�xj

�D

�xiuj

�ui�D

�xj� ���ijrD � u

��

dx

and �A�� allows to estimate the �rst term in S� by

T� � ��ZIRn����eA��De�Ajuj�dx�

All in all we have

R� R� � R� S� S� � �R� S� T�� T�

�ZIRn

��IV�eA��D�juj� �����eA����D�u�

�u

�xu �Djuj�u � rD�

�����eA��Xi�j

�D�ui�xj

�D

�xiuj

�ui�D

�xj� ���ijrD � u

�� e�Adx

��

ZIRn

����eA��De�Ajuj�dx�

The �rst integral can be written asZIRn

tr�XY �e�Adx�

where X and Y are the �� ��matrices

X �

������eA�� ������eA��������eA�� �IV�eA��

�and� resp��

Y �

�� D�u� �u

�xu �

�Djuj�u � rD

D�u� �u�xu �

�Djuj�u � rD D�juj�

��

with

� �Xi�j

�D�ui�xj

�D

�xiuj

�ui�D

�xj� ���ijrD � u

��

X is non�negative de�nite since � generates an admissible entropy �cf� De�nition����� A simple calculation shows that Y is also nonnegative de�nite� Thus�Z

IRntr�XY �e�Adx �

��

and we have for the entropy production rate �������

R����t�j��� � ��ZIRn

����eA��De�Ajuj�dx � ���I����t�j����

The assertion now follows from

d

dtjI����t�j���j � ���jI����t�j���j� ������

In a special case of equation ����a� the condition �A�� has a simple geometricinterpretation�

Remark ���� For D�x� � I condition �A�� simply requires the uniform conve�xity of A�x� on IRn i�e�

�A�� �� such that��A�x��xi�xj

�i�j����� �n

� �I x � IRn�

The condition �A�� is a special case of the well�known Bakry�Emery condition forlogarithmic Sobolev�inequalities �BaEm���� �Bak���� �Bak���� In fact� the proof ofLemma ��� is a concretization of the approach of Bakry and Emery and was givenhere mainly for the sake of clarity� For general �symmetric and uniformly positivede�nite� di�usion matrices D�x� an �excursion� into basic di�erential geometryis� however� in order to understand the Bakry�Emery condition� Therefore weconsider the Riemannian manifold corresponding to the di�usion matrix D�x� ��dij�x�� as metric tensor� The Christo�el symbols are de�ned as the elements ofthe ��tensor�

"lij �nX

k�

�dkl��dij�xi

�dki�xj

� �dij�xk

�� ������

where dkl�x� are the entries of the inverse D�x��� � �dkl�x��� The Riemanncurvature tensor then reads

Rkilj �

�xi"ljk �

�xj"lik

nXm�

"lim"mjk �

nXm�

"ljm"mik ��� �

and the Ricci�tensor

�ij �nX

k�

Rikkj � ��� ��

��

The covariant derivative of a vector �eld X � �X�� � � � � Xn� is given by

riXj ��Xj

�xi

nXk�

"jikXk� ��� ��

We de�ne the symmetric ��tensor

�rSX�ij��

nXl�

�djlriXl dilrjX

l�� ��� ��

The Ricci tensor of the Fokker�Planck operator is de�ned as

Ricij�x� �nX

k�l�

dikdjl��kl �

�rSX�kl�x��

��� �a�

where we set

X i�x� � �nXj�

dij�

�xj

�A�x�� �

�ln detD�x�

�� ��� �b�

Then the Bakry�Emery condition for a general symmetric positive de�nite di�u�sion matrix reads

�A�� �� such that D�x�Ric�x� � �I x � IRn�

For the case D�x� � I we shall �rst compare �A��� or rather �A��� to the corre�sponding condition given in Prop� ��� of �Bak���� If D � I� their condition

mX�x��X�x� � m� n

m�Ric�x�� �D�x����

simpli�es to

mrA�rA � m� n

m

���A

�x�� �I

� x � IRn� ��� �

with some � � IR and m � �n��� denoting� respectively� the curvature anddimension of the FokkerPlanck operator L� If we assume � and m to be constant��� � only admits a global solution A�x�� x � IRn if m � �� Hence� ��� �reduces to �A�� with � � ��

We have

Lemma ����� If A � A�x��D � D�x� satisfy �A�� then the estimate ������holds�

��

Proof� See �BaEm���� �Bak����

Remark ����� To better understand the Bakry�Emery condition �A�� we consi�der a transformation of the Fokker�Planck type equation under an x � y di�eo�morphism

y � y�x� �� x � x�y�

on IRn� We denote J�y� � det �x�y�y� assume J on IRn and set

���y� t� � J�y���x�y�� t��

A lengthy calculation �cf� also �Ris���� gives

��t � divy�eD�ry�� ry� 'A� lnJ����� ��� �a�

where

eD�y� � �y�x�x�y��D�x�y��

��y

�x�x�y��

��� ��� �b�

'A�y� � A�x�y��� ��� �c�

Assume now that the Riemann curvature tensor ������ vanishes� Then the geo�metry produced by D is Euclidean and there exists a transformation y � y�x�

such that eD�y� � I �Ris���� The condition �A�� applied to ����a� reduces to �A��applied to ���� a� i�e� we need to assume the uniform convexity of 'A � lnJ inthe y�coordinates�

In one dimension �n � �� the Riemann curvature tensor always vanishes and thetransformation which yields a Fokker�Planck equation with di�usion coe�cient �can be constructed explicitly� It is de�ned by

dy

dx�x� �

pD�x�

��� � x � IR� ��� ��

For arbitrary n � an analogous transformation works if D is a constant matrix�We set

�y

�x�x� �

pD���� x � IRn� ��� ��

Then the transformed equation has the identity matrix as di�usion matrix�

In general �i�e� for a non�vanishing Riemann curvature tensor� the metric tensorcannot be transformed to the identity by a coordinate transformation� It can betransformed to the form eD�y� � I eD�y� where eD is a scalar function if isothermalcoordinates exist on the manifold corresponding to D� Locally at least this isalways the case for n � � �cf� �BeGo��� p������

��

From the exponential decay of the entropy production I� �Lemma ���� we shallnow derive the exponential decay of the relative entropy e��

Theorem ����� Let e� be an admissible relative entropy and assume thate���I j��� �� Let the coe�cients A�x� and D�x� satisfy condition �A��� Thenthe relative entropy converges to � exponentially�

e����t�j��� � e����te���I j���� t � ��� ��

Proof� We proceed in two steps and �rst derive ��� �� for �I � S �� f� �L���IR

n� ���� �dx�� I���j��� �g�

From the Lemmata ���� ���� we then know that e����t�j���� and I����t�j���� as t ��� Hence� integrating ������ �which also holds under condition �A�� �see �BaEm���� �Bak���� over �t��� gives

I��t� �d

dte��t� � ���e��t�� �����

which proves the assertion for su!ciently regular initial data�

For the general case we use a density argument to approximate �I in two steps� ��is given in L���IR

n�� and �I is a normalized �i�e�R�Idx �

R��dx � �� L���IR

n��

function with �nite relative entropy� �I � f� � L���IRn�e���j��� �g�

We �rst approximate �I by �N � L���IRn� withR�Ndx � ��N � IN��

�N�x� �� �N �I�x��lf�I����Ng

with the monotonously decreasing normalization constants �N �

� Rf�I����Ng

�Idx

��� � for N��� By construction we have j �N

��j � N�N � which implies �N �

L��IRn� ���� �dx���

�N converges to �I a�e�� and also in L��IRn��

k�I � �NkL��IRn� � j�� �N jZf�I����Ng

�Idx

Zf�I����Ng

�IdxN���� �

Now we construct a uniform bound for �� �N������ N � IN � we consider the convex�

monotonously increasing function �R���� � de�ned by �R��� � ��R� for � � R and �R��� � ���� for R �� with R � su!ciently large such that�R � � on IR�� This implies

�R��� � ���� ��R�� � � ������

From Lemma ���b we easily conclude with �� ��

�R���� � ��R��� ����� � � � ������

Since �N � � we have �N � ��I �for large N� and we estimate using ��������������

���N����� � �R��N

����� ������

� �R���I����� � ��R� �I

����� ����I � ��� �I

����� ���R��� ����I �� �����

Our assumptions on �I showRIRn����d� �� and Lebesgue�s dominated con�

vergence theorem gives

limN��

e���N j��� � e���I j���� ������

In the second approximation step we shall approximate �N in L���IR

n� ���� �dx��by the normalized �i�e�

R�N�Mdx � �� sequence f�N�MgM�IN � having �nite

entropy production� We choose �N�M � C����IRn� which denotes non�negativeC��functions with compact support in IRn�

We shall now show that

C����IRn� � S � fp� � L���IRn� ���� �dx��

jI���j���j �g� ���� �

We consider � withp� � C����IRn� and compact support () � supp� � IRn� Since

A � L�loc�IRn�� so is ��p�� � eA�� and hence ��

p�� � L��IRn� follows�

Using

������ � ���� ���� � �

which follows from ������� we estimate the entropy production �������

jI���j���j �Z�

�����

���uTD�x�u��dx

� ��kDkL����

Z�

�� ����jr� �

���j���dx

� ���kDkL����

Z�

�� ���

rr �

��

� dx� ���kDkL����

Z�

�� �

�����jrp�j� �jrAj��dx� ������

��

which is �nite since D�x�� ���� �x� and rA�x� � L�loc�IRn�� Hencep� � S�

Since C�� �IRn� is dense in L��IRn� ���� �dx��� �N can indeed be approximated by

f�N�Mg � S in L��IRn� ���� �dx��� Lemma �� then shows

limM��

e���N�M j��� � e���N j���� ������

From these two approximations we extract a normalized �diagonal� sequencef�N�M�N�g � S with

�N�M�N�N���� �I in L

��IRn�� �����a�

limN��

e���N�M�N�j��� � e���I j���� �����b�

For the approximations �N�M�N� we can apply ��� ���

e���N�M�N��t�j��� � e����te���N�M�N�j���� t � ������

where �N�M�N��t� denotes the solution of ����a� with initial data �N�M�N��

From the entropy bound ������ and from the Dunford�Pettis theorem �Ger��� weeasily conclude

�N�M�N��t�

��N���� ��t�

��in L��IRn� ���dx�� weakly�

We now use the lower semi�continuity of e���j��� to �nish the proof�ZIRn�

���t�

��

����dx� � lim inf

N��

ZIRn

��N�M�N��t�

��

����dx�

� e����te���I j���� t �

Due to the above density argument we do not have to make the #algebra hy�pothesis$ of �BaEm��� and x� of �Bak���� there one assumes that there exists acore for L that is stable under the evolution eLt and under composition with C�

functions� Using di�erent techniques such a density argument was also given inx� of �DeSt����

��

��� Convex Sobolev inequalities in entropy version

����� is the entropy version of a convex Sobolev inequalityZIRn

��

��

����dx� � �

��

ZIRn

�����

��

�r�

��

��

�Dr

��

��

����dx�

����a�

� � L���IRn� with

ZIRn

�dx �

ZIRn

��dx� ����b�

This inequality� of course� does not require our usual normalizationR��x�dx � ��

The relation of ����� to other versions of this inequality will be discussed in x��Note that L���IR

n� in ����b� can be replaced by L��IRn� if � is quadratic�

The desired L��convergence of ��t� to �� is now a direct consequence of Theorem���� and the Csisz�ar�Kullback inequality ������� �������

Corollary ����� Let e� be an admissible relative entropy and assume thate���I j��� �� Let the coe�cients A�x� and D�x� satisfy condition �A��� Thenthe solution of ����� satis�es

k��t�� ��kL��IRn� � e���tr�

��e���I j���� t � ������

with the notation �� � �������

To complete the line of argumentation we shall now show that the HamiltonianHhas a spectral gap if A andD satisfy �A��� This condition implies the logarithmicSobolev inequalities ����� for any admissible relative entropy e�� We start byrewriting the Sobolev�inequality ����� for ���� � � ln� � � � by settingr

���

jgjkgkL��IRn����dx��

�cf� also x��� We obtain after a simple calculationZIRn

jgj� ln jgj���dx� � �

ZIRn

rg�Drg���dx� kgk�L��IRn����dx�� ln kgkL��IRn����dx��

������

for all g � L��IRn� ���dx��� The well�known Rothaus�Simon mass gap theorem��Rot���� �Sim���� �Gro���� givesZ

IRn

rg�Drg���dx� � �kgk�L��IRn����dx�� �����a�

��

if ZIRn

g���dx� � � �����b�

Simple calculations show that

H ��p��Lp��� � DQ � L��IRn� dx�� L��IRn� dx�

where H is given by ����� andZIRn

rg�Drg���dx� �ZIRn

p��gH�

p��g�dx�

Using the spectral theorem it is easy to conclude that the spectral gap � of Hsatis�es � � �� i�e��

��H� �� �� � ��In many cases the logarithmic Sobolev constant � is indeed smaller than thespectral gap � �see� e�g�� example ������ x� of �DiSC���� and the discussions in�Rot���� �Bak���� �DeSt����

Remark ����� Assume that D�x� is pointwise in IRn bounded below by a sym�metric positive de�nite matrix D��x� i�e�

D��x� � D�x�� x � IRn�in the sense of positive�de�nite matrices� and that the Fokker�Planck operator

L���� �� div�D��r� �rA��satis�es the Bakry�Emery condition �A�� �with D replaced by D��� Then theconvex Sobolev inequality ������ holds with D replaced by D�� Since ��� � wealso have Z

IRn�����

��

�r�

��

��

�D�r

��

��

����dx�

�ZIRn

�����

��

�r�

��

��

�Dr

��

��

����dx� ������

and ������ follows� Consequently the decay statement in Lemma ���� Theorem���� and Corollary ���� and the above mass gap argument hold for L�

Note that this settles the case

D�x�I � D�x�� x � IRn�where D�x� and A�x� satisfy �A��� In particular for a uniformly positive de�nitedi�usion matrix

dI � D�x�� x � IRn�with d � IR�� and a uniformly convex potential A ��A�� holds� we obtain theSobolev inequality ������ where ������ has to be replaced by ����d���

��

��� Non�symmetric Fokker�Planck equations

At the end of this Section we extend the above analysis to the following class ofFokker�Planck type equations with certain rotational contributions to the drift�coe!cient� We consider

�t � div�D�r� ��rA �F ���� x � IRn� t � ���� �

��t � � � �I � ������

with the above conditions on �I � D and A� Additionally we assume for �F ��F �x� t� su!cient regularity and

div�D�F��� � on IRn � ����� ������

such that �� � e�A is still a stationary state of ���� �� Our objective is again toanalyze the rate of convergence of ��t� towards ��� For the symmetric problem����� we proceeded in x��� by deriving a di�erential inequality for the entropyproduction �see �������� In contrast� we shall here only apply the convex Sobolevinequality ����� that was obtained for the corresponding symmetric problem �i�e�

���� � with �F � ��

With the transformation � � ���� �cp� ������� ���� � may be rewritten as

�t � ���� div�D��r�� �F�Dr��

where the second term of the r�h�s� is skew�symmetric in L��IRn� d����

We �rst calculate�

d

dte����t�j��� � I����t�j��� T�

with I� as in ���� � and

T �

ZIRn

����

��

�div�D�F��dx�

We use ������ in the form

div�D�F �� � ��D�F �r�� ���

to obtain

T �

ZIRn

����

��

��D�F �r

��

��

���dx �

ZIRn

r����

��

�D�F��dx�

An integration by parts �nally gives

T � �ZIRn

��

��

�div�D�F���dx � �

By the convex Sobolev inequality ����� we obtain

d

dte����t�j��� � ���e����t�j���

and

e����t�j��� � e����te���I j���

follows again�

As an example of ���� � we consider the kinetic Fokker�Planck�type equation forthe distribution function f�x� v� t��

ft fA� fg � div�x�v��e�A�x�v�rx�v�eA�x�v�f��� t � �����a�

f�t � � � fI � L���IR�n�� �����b�

with the position variable x � IRn and the velocity variable v � IRn� Here

fA� fg �� rvA � rxf �rxA � rvf

denotes the Poisson bracket� For the sake of simplicity we set D � I� With thechoice �F � �rvA �rxA�

� the convergence of f�t� to its steady state f��x� v� �e�A�x�v� follows from the above calculations�

Theorem ����� Let fI � L���IR�n�� A � W ���loc �IR

�n� and � �RIR�n fI�x� v�dxdv �R

IR�n e�A�x�v�dxdv� Let e� be an admissible relative entropy and assume thate��fI jf�� �� Let A�x� v� be strictly convex i�e�

�� such that��A

��x� v��� �I x� v � IRn�

Then the relative entropy converges to � exponentially�

e��f�t�jf�� � e����te��fI jf��� t � ������

��

� Sobolev Inequalities

��� Three versions of convex Sobolev inequalities

We shall now discuss in detail the convex Sobolev inequality ������ In particularwe shall rewrite ����� in various equivalent forms and discuss its relation to otherknown inequalities� At �rst we set u � ����� Then ����� becomesZ

IRn

��u����dx� � �

��

ZIRn

����u�r�uDru���dx� ����a�

for all u � L���IRn� �L��IRn� if � is quadratic� which satisfyZIRnu���dx� �

ZIRn

���dx�� ����b�

Assume for the followingRIRn���dx� � ��

Hence setting u � v�RIRnv���dx�� we obtainZ

IRn

�vR

IRnv���dx�

����dx� � �

��

ZIRn

����

vRIRnv���dx�

� r�vDrv�RIRnv���dx���

���dx�

�����

for all v � L���d��� �� L���IRn� ���dx�� �v � L��d��� if � is quadratic��The most common form of convex Sobolev inequalities is obtained by settingv � f � in ������ This gives the so called steady state measure version�ZIRn

�f �

kfk�L��d���

����dx� � �

ZIRn

f �

kfk�L��d���

����

f �

kfk�L��d���

�r�fDrf ���dx�

�����

for all f � L��d����The inequalities ����������� hold for all functions �� � e�A �with L��dx��normequal ��� �� on IR

n and symmetric positive de�nite matrices D � D�x��which are su!ciently smooth �cf� Section �� and which satisfy �A�� �if D�x� �D�x�I�� or �A�� �if D�x� � I�� or �A��� or there exists a symmetric positivede�nite matrix D� � D��x� � D�x� such that A and D� satisfy �A���

If D�x� � I� ���x� �Ma�x� ���

���a�n��exp�� jxj�

�a� for some a then �A�� holds

with � � ��a and we obtain the celebrated Gross logarithmic Sobolev inequality�Gro� �� Z

IRn

f � ln

�f �

kfk�L��dMa�

�Ma�dx� � �a

ZIRn

jrf j�Ma�dx� �����

��

for all f � L��dMa�� where we set ���� � � ln� � � � �see also ��������A second example is provided by the same choices of �� and D setting ���� ��� � ���� We then haveZ

IRn

v�Ma�dx���Z

IRn

vMa�dx�

��

� aZIRn

jrvj�Ma�dx� ��� �

for all v � L��dMa�� This is an old inequality� and as remarked by Beckner �Bec���in one dimension was probably known to both mathematicians and physicists inthe ����s �Wey���� It has been a useful tool in di�erent subjects� like partialdi�erential equations �Nas �� and statistics �Che����

Finally let � � �p��� � �p � � � p�� � ��� see �����b�� Then we obtain the

inequalityZIRn

vpMa�dx���Z

IRnvMa�dx�

�p

� �ap� �p

ZIRn

jr�vp���j�Ma�dx� �����

for all v � L���dMa�� � p � �v � L��dMa� for p � ��� Setting juj � vp�� givesthe generalized Poincar�e�type inequality by Beckner �Bec����

p

p� ��Z

IRn

u�Ma�dx���Z

IRn

juj��pMa�dx�

�p�� �a

ZIRn

jruj�Ma�dx� �����

for all u � L��p�dMa�� � p � ��We remark that the inequalities ����� interpolate in a very sharp way between thePoincar�e�type inequality ��� � and the logarithmic Sobolev inequality ������ whichis obtained from ����� in the p� � limit� ����� and ����� represent a hierarchy ofconvex Sobolev inequalities� with the logarithmic Sobolev inequality ����� beingthe �strongest�� This� however� cannot be seen directly from ������ ����� andrequires a more involved line of argument �see �Bak���� p� �� where the spectralgap inequality ��� � is compared to the logarithmic Sobolev inequality ������� In�Led��� this interpolation is discussed for the Ornstein�Uhlenbeck process on IRn

and for the heat semigroup on spheres�

Note that the inequalities ������ ��� �� ������ ����� also hold when the Gaussianmeasure Ma is replaced by a general steady state �� � e�A� such that �A�D�satis�es the Bakry�Emery condition �A��� The quadratic form on the r�h�s� isthen replaced by �

��

R ru�Dru���dx��Let us now discuss brie*y some consequences of inequalities ������ ��� �� ������������

In spite of the fact that the class of admissible entropies and steady state measureswhich generate inequalities ������ ��� �� ������ ����� is very wide� Sobolev inequa�lities with respect to the Lebesgue measure follow only if ���� � � ln� � � ��

��

Actually� in this case the choice g� � f �Ma leads to the inequalityZIRn

g� ln

�g�

kgk�L��dx�

�dx

n n

�ln ��a

�kgk�L��dx� � �a

ZIRn

jrgj� dx �����

for all g � L��IRn�� a �cf� �Car����� Inequality ����� is the logarithmic Sobo�lev inequality for the heat kernel H� � �&� In Davies� book �Dav��� inequality����� is contained in the more restrictive Sobolev inequalities framework of ultra�contractive operators �see Theorem ����� and the rest of the Chapter�� To obtainDavies� form� we rewrite ����� as a family of logarithmic Sobolev inequalities for� �Z

IRn

g� ln g dx � �ZIRn

jrgj� dx MG���kgk�L��dx� kgk�L��dx� ln kgkL��dx� �����

for all � g � L��IRn�� As will be shown in Theorem ��� below MG��� ���

�n n

�ln ���

�is the sharp constant for all � � Now� let us compare the

function MG of ����� with the analogous one given by Davies� conditions� In hisframework� inequality ����� follows if��e�H�tg

��L��dx�

� kgkL��dx�eM�t�� t � �����

whereM�t� is a monotonically decreasing continuous function of t� In the presentcase� H� � �&�

e�H�tg � Kt � g� ������

where Kt�x� is the Gaussian ���t��n��exp

n� jxj�

�t

o� Then Young�s inequality gives��e�H�tg

��L��dx�

� kgkL��dx�kKtkL��dx� ������

and we obtain M�t� � �n�ln ��t� Thus

M����MG��� �n

���� ln �� � ������

and the function M��� is not optimal�

In the above example we were able to rewrite ����� as ����� with arbitrarily smallprincipal coe�cient �� For general potentials A�x� that satisfy �A��� the possibi�lity of doing so is deeply connected to the di�erence between hypercontractivityand ultracontractivity �see x of �Gro���� x� of �Dav����� As the heat kernelexample shows� the entropy approach of x� could lead to better constants�

��

��� Perturbation lemmata for the potential A�x�

In Section � we derived the convex Sobolev inequality ����� corresponding toFokker�Planck operators L� � �div�D�r� �rA�� that satisfy the Bakry�Emerycondition �A��� Next we will present two perturbation results to extend thisinequality to a larger class of operators�

First we shall consider bounded perturbations of the �potential� A�x�� Our resultgeneralizes the perturbation lemma of Holley and Stroock �HolSto���� �Gro��from the logarithmic entropy to all admissible relative entropies e� from De�ni�tion ����

Theorem ���� Let ���x� � e�A�x�� f���x� � e� eA�x� � L���IRn� with RIRn��dx �RIRnf��dx � � and

eA�x� � A�x� v�x��

a � e�v�x� � b �� x � IRn� ������

Let the symmetric locally uniformly positive de�nite matrix D�x� be such that theconvex Sobolev inequality ����� �with the admissible entropy generator �� holdsfor all f � L��d����Then a convex Sobolev inequality also holds for the perturbed measure f�� �Z

IRn

�f �

kfk�L��df���

� f���dx� ���� �

� �

�max�

b

a��b�

a�

ZIRn

f �

kfk�L��df���

����

f �

kfk�L��df���

�r�fDrf f���dx�

for all f � L��df��� � L��d����Proof� We introduce the notations

���x� ��f ��x�

kfk�L��d���

� ���x� ��f ��x�

kfk�L��df���

� � �������kfk�L��d���

kfk�L��df���

and because of ������ we have

b� � � �

a� ������

Below we shall need the estimate

������� ��

�������� �� ��� �����a���������� � �� � ��� �����b�

The �rst line follows from ���� � �see ������� and the second line follows from�������

We shall now �rst prove the assertion ���� � for the case � � �� In the followingchain of estimates we use� in this sequence� ������� ������� ������ ������� �����b���������Z

IRn

�����f���dx� �ZIRn

�������

� ����� ���� � ��� f���dx�

� ��ZIRn

�����f���dx� � b��ZIRn

��������dx�

� b���

ZIRn

f �

kfk�L��d���

�������r�fDrf���dx� ������

� b

a

ZIRn

f �

kfk�L��df���

�������r�fDrff���dx�� b

a

��

ZIRn

f �

kfk�L��df���

�������r�fDrff���dx�� b

a��

ZIRn

f �

kfk�L��df���

�������r�fDrff���dx��which is the result if �� � ���In the case � � we proceed similarly and �rst apply ������ to obtain�ZIRn

�����f���dx� � ZIRn

������� ����� ���� � ���f���dx� � �ZIRn

�����f���dx��������

Now we again use ������� ������ ������� �����a�� ������ and �nally obtain the resultZIRn

�����f���dx� � b�a

ZIRn

f �

kfk�L��df���

�������r�fDrff���dx�� �����

We remark that the �perturbation constant� in ���� � is not as good as the onein the proof of �HolSto��� �max� b

a�� b

a� versus b

a�� This is due to the fact that the

proof of Holley and Stroock exploits homogeneity properties of � and ��� in thecase ���� � � ln��� �� Thus� their original proof �with the constant b

a� can be

extended to entropy generators of the form ���� � �p� �� p��� ��� � p � ��but not to the general case of Theorem ����

Example ���� Set D�x� � I and consider the �double well potential� A�x� �c�jxj�� c�jxj��c��� �� Then the perturbation theorem ��� yields convex Sobolevinequalities as A�x� can be written as a bounded perturbation of a uniformlyconvex potential that satis�es �A���

��

Next we shall specialize our discussion to the one�dimensional situation and deriveconvex Sobolev inequalities under very mild assumptions on the potential A�x��In particular we shall prove that an appropriate choice of the di�usion coe!cientD compensates lack of convexity of the potential A�

Theorem ���� Let A � W ���loc �IR� be bounded below and satisfy ���x� � e�A�x� �

L���IR� withRIR���dx� � �� and let � generate an admissible relative entropy�

Then there exists a � and a function D � D�x� with D�x� � D�� x � IR�for some constant D� � such that�Z

IR

��

��

����dx� � �

��

ZIR

�����

��

�D

� ����x

� ���dx�� ������

� � L���IR� withZIR

�dx � ��

The idea of the proof is to construct a bounded function D�x� such that �A�D�satis�es �A��� To this end we need the following

Lemma ���� Let A�x� satisfy the conditions of Theorem ���� Then there existsa such that the ODE

D�x

D� ��Dxx

�DxAx DAxx � ������

admits at least one global solution that satis�es D�x� � D� � x � IR�

Note that the left hand side of ������ is the �� d version of the left hand side of�A�� such that Theorem ��� follows�

Proof� We transform D�x� � y�x�� and solve

yxx � Axxy Axyx � y� x � IR� ������

y�� � eA���� y��� � Ax��eA���� ���� �

The �possibly only local� solution of ������ satis�es

y�x� � eA�x� � eA�x�Z x

�e�A�z�

Z z

y�����d��dz � eA�x�� ������

Due to this upper bound� y�x� can only break down at a �nite x� if y�x�� �� Locally� y�x� can be obtained through a �xed point iteration starting withy��x� �� e

A�x� �

yk���x� � eA�x� � eA�x�Z x

�e�A�z�

Z z

yk�����d�

�dz ������

� eA�x���� ke�AkL��IR�

Z x

yk�����d�

� � k � �

��

Starting with y��x� � eA�x�� an iterative application of the estimate ������ yields

y�x� � �eA�x� ������

with

� � ��

��

�� � � �

��

r�

�� �

whenever � ���

By induction one shows that �yk� is decreasing� Hence� ������ converges toy�x� x � IR� From ������ we obtain D�x� � � �e�A� �� D�� x � IR� withA� denoting the lower bound of A�x��

Note that D � e�A satis�es ������ with � � Hence the couple �A�D � e�A�violates the Bakry�Emery condition �A��� nevertheless the convex Sobolev ine�quality ������ holds with D � e�A and � �

���due to the estimate ��������

Obviously� for A uniformly convex D can be chosen as a constant �cf� �A����We shall now show that for nonuniformly convex A� in general �A�� cannot besatis�ed with a uniformly bounded function D�

Lemma ���� Let A�x� satisfy the conditions of Theorem �����

a� If D�x� � D� �� x � IR� then limx��Ax�x� � ���b� If D� � D�x� � D� �� x � IR� then A�x� grows at least quadrati�

cally�

Proof� a� Using D � y� we rewrite the di�erential inequality �A�� as

yxx

y f � �Axy�x

for some f�x� � � Solving for A gives

Ax�x� ��

y�x�

�C

Z x

dz

y�z�

Z x

f�z�dz yx�x�

�� ������

for some C � IR� Since y is bounded on IR we have limx��yx�x� ��� Theresult follows from Z x

dz

y�z�� xp

D�

� x � �

��

b� Integrating ������ gives

A�x� � A��

�Z x

dz

y�z�

��

C

Z x

dz

y�z� ln

y�x�

y��

Z x

y�z�

Z z

f���d�dz�

and the result follows from the boundedness assumptions on D�

We now �nish our discussion of perturbation results on convex Sobolev inequali�ties with an example� It demonstrates that the di�usion coe!cient D constructedin the proof of Lemma ��� in many cases has a much stronger growth at x � ��than necessary to satisfy the Bakry�Emery condition �A���

Example ��� Let A � C��IR� satisfy A�x� � cjxj�� for jxj L and � � ��Then a pair �D� � satisfying �A�� can be constructed such that

D�x� �

� �c�� c

�� x

����� � x L��c�� c

�� jxj���

��� x �L�

� �����

for some c� � IR c� L� L� The construction of D proceeds as follows�On the interval ��L�� L��� where L� will be determined later we choose D � y��where y is the solution ���� � of the IVP ������� Outside of this interval weextend D by ������ in a C��way thus �xing c��� �in dependence of and L��� Bya perturbation argument around � one easily sees that can be chosensmall enough and L� large enough such that �yx�x���L�� � and such that �A��holds for jxj L��

��� Poincare�type inequalities

As an application of Theorem ��� we shall now derive Poincar�e�type inequalities�

Remark ���� Poincar�e�type inequalities on bounded uniformly convex domainsare readily obtained from convex Sobolev inequalities� For simplicity�s sake letB � fx � IRnj jxj �g be the unit ball in IRn and let �� � ���x� � � ���x� � �on B be the density of a probability measure on B� We de�ne the C��function'A� � 'A��jxj� on IRn for � by

d�

dr�'A��r� �

��� r ���� r �

� 'A��� �d

dr'A��� � �

We set

A��x� �

�'A��jxj�� jxj�

�� ln ���x�� x � B

'A��jxj�� x �� B

��

and

����x� � exp��A��x���ZIRn

exp��A��y��dy�

Obviously

����x������

����x�� x � B� x �� B �

Since A� �� � ln ��� is an L�� perturbation �uniformly as �� � of the uniformlyconvex function 'A��jxj� which satis�es

�� 'A�

�x�� I on IRn�

we can apply the perturbation result Theorem ��� and obtain the convex Sobolevinequality �with D � I�Z

IRn

�vR

IRnvd���

�d��� � c

ZIRn

����

vRIRnvd���

� jrvj��RIRnvd�����

d���

for all admissible entropies � and all functions v � L���IRn� d���� �v � L��IRn� d����if � is quadratic on IR�� Here c is independent of ��

Passing to the limit �� gives the Poincar�e�type inequalityZB

�vR

Bvd��

�d�� � c

ZB

����

vRBvd��

� jrvj��RBvd����

d��

for all v � L���B� d��� �v � L��B� d��� if � is quadratic on IR�� In the latter case���� � �� � ��� with �� � �

vol�B�we obtain the classical Poincar�e inequalityZ

B

�v � �

vol�B�

ZB

vdx

��

dx � �cZB

jrvj�dx� v � L��B��

Obviously the above limit argument can easily be carried over to general uniformlyconvex domains�

��� Sharpness results

We now turn to the analysis of the saturation of the convex Sobolev inequalities�i�e� we shall answer the question for which function � the inequality ����� becomesan equality� In particular we shall �nd necessary and su!cient conditions onthe entropy generator � and on ��� which imply the existence of an admissible

function � � �� such that ����� �under the assumption D � I� becomes anequality� We remark that this question was completely answered in �Car��� forthe Gross logarithmic Sobolev inequality �i�e� ���� � � ln� � � �� �� � Ma�by a technique di�erent from the one presented in the sequel� The same problemwas treated in �Tos��a� and �Led��� using a method which we shall generalizebelow�

At �rst we observe that the derivation of the convex Sobolev inequalities givenin Section � is based on writing the entropy equation

d

dte����t�j��� � I��e�t�j���� ������

the equation for the entropy production

d

dtI����t�j��� � ���I����t�je�� r����t�� ������

and proving r����t�� � � Explicitly� we obtain by inserting ������ into the righthand side of ������ and by integrating with respect to t�

e���I j��� � � �

��I���I j���� �

��

Z �

r����s��ds ������

where ��t� in the Fokker�Planck trajectory which �connects� the initial state �Iwith the steady state ��� r� � then gives the convex Sobolev inequality

e���I j��� � �

��jI���I j���j�

which becomes an equality i�Z �

r����s��ds � �� r����t�� � a�e� in IR�t � ������

The precise form of the remainder r� can easily be extracted from the proof ofLemma ���� Assuming henceforth

D � I ���� �

we obtain

r���� �

ZIRn

��IV�eA��juj� ������eA��u��u

�xu �����eA��

nXl�m�

��ul�xm

��

�e�Adx

ZIRn

����eA��u����A

�x�� �I

�u e�Adx� ������

��

where we recall u � r�eA��� Obviously� r���� � if u � � which �takinginto account the normalization� implies � � �� and gives the trivial case ofequality in the convex Sobolev inequality� Thus� assume u � � Then� using theadmissibility conditions ������ for the entropy generator � and �A��� we concludethat r���� � holds i� the subsequent four conditions are satis�ed�

�����eA��� ��

�����eA���IV�eA��� �����a�

juj��

nXl�m�

��ul�xm

��

� ��

� ju��u�xuj� �����b�

����eA��u��u

�xu � ������eA��juj�� �����c�

u����A

�x�� �I

�u � � �����d�

Since ��t� �� eA��t� satis�es ������ we have by the maximum�minimum principle

infIRneA�I � eA�x���x� t� � sup

IRneA�I

a�e� in IRn � IR�t � We conclude from �����a��

Lemma ���� If the Sobolev inequality ������ with D � I becomes an equality for� � �� then ���� � ���� or ���� � ���� holds for � � �infIRn eA�I � supIRn eA�I�where � and � are given by ���� a� and ���� c� resp�

Nontrivial saturation can therefore only occur for the #minimal$ and #maximal$entropies�

At �rst we investigate the case of the maximal �quadratic� entropy� Note thatnow any � � L��IRn� dx� is admissible� Positivity is not required�Theorem ���� The convex Sobolev inequality ������ with D � I and ���� ��� � ��� becomes an equality i� the following two conditions hold�

�i� there exist Cartesian coordinates y � �y�� � � � � yn�� � y�x� on IRn such that

for some � � IR

A�x�y�� ���y�� �y� B�y�� � � � � yn�� ������

��

�ii� � satis�es for some � � IR�

��x�y�� � �� �y��e�A�x�y��� ������

Proof� Since ���� � we conclude from �����b� and �����c���ul�xm

� % l� m � �� � � � � n �����

�u � +�� Thus u�x� t� � rx�eA�x���x� t�� � C�t�� where C�t� is constant in

x� Then ��x� t� � �C�t� � x C��t�� e�A�x� follows with C� real valued� Inserting

into the Fokker�Planck equation gives ,C � x ,C� � �rA � C and ,C � ���A�x�C

follows� �����d� gives C�t����A�x�

� �I�C�t� � and since ��A

�x�� �I �

we conclude ��A�x�C�t� � �C�t�� We obtain ,C � ��C and C�t� � C�e

���t�

Therefore��A�x�

� �I�C� � and we have rA�C���C� �x � � � IR� We insert

C�t� � C�e���t into ,C �x ,C� � �rA �C and �nd C��t� �

���e���t C�� C� � IR�

This gives

��x� t� �

�C� � x �

�e��t�A�x� C�e

�A�x�

and t�� implies C� � �� Summing up� we found� for some C� � IRn� � � IR�

��x� t� �

�C� � x �

�e���t�A�x� e�A�x� �����a�

and

r�A� ��jxj�� � C� � �� �����b�

Note that C� � implies � � and ��x� t� � e�A�x� follows� We therefore assumeC� � from now on�Set �� �

C�

jC�j � choose orthonormed vectors ��� � � � � �n � f��g and de�ne thechange of coordinates x � y by x � y��� � � � yn�n� We have rxf�x� � C� ��f�x�y���y�

jC�j and thus

A�x�y�� ���y��

jC�jy� B�y�� � � � � yn�

follows from �����b�� and this �nishes the proof�

Next we treat the #minimal$ entropy case�

��

Theorem ����� The convex Sobolev inequality ������ with D � I and ���� �� ln� � � � becomes an equality i� the following two conditions hold�

�i� there exist Cartesian coordinates y � �y�� � � � � yn� � y�x� on IRn such that

for some � � IR

A�x�y�� ���y�� �y� B�y�� � � � � yn��

�ii� � satis�es for some � � IR

� � exp

��A�x�y�� �y� � �

�� ��

�� ������

Proof� We set z � r ln��eA� in ������ and calculate �using the speci�c form of���

r���� � �

ZIRn

nXl�m�

��zl�xm

��

dx �

ZIRn

�z����A

�x�� I

�zdx�

Assume z � � Then �z�x� follows and z�x� t� � C�t� � IRn� This gives��x� t� � C��t� exp�C�t� � x� A�x��

with C��t� � Inserting into the Fokker�Planck equation and proceeding as inthe proof of the previous Theorem implies

��x� t� � exp�C� � xe��t �

�e��t � A�x�

�������

where � and C� satisfy �����b�� Setting t � in ������ proves the assertion�

Note that � in ������ is obtained by a shift in the y��coordinate of �� �cf� �Car�����

In one dimension �n � �� or for constant matrices D the analogous result tothe Theorems ��� and ��� is obtained by the coordinate transformation of Re�mark ���� For D�x� � I� however� nontrivial saturation of the convex Sobolevinequality ����� is not always possible for any A�x�� even for scalar di�usionD�x� � D�x�I� the analogue of ����� implies an integrability condition on D�x��which is not satis�ed in general�

� A generalized Csisz�ar�Kullback Inequality

We shall now be concerned with proving generalized Csisz�ar�Kullback inequali�ties� which are estimates for the L��distance of two functions in terms of their rela�tive entropy� These inequalities have at least a � years history in probability and

��

information theory �Csi��� Csi��� Csi��� Kul �� KuLei �� Per �� Per� � BaNi����In �Csi��� �Theorem ���� page ��� and Section �� page ���� the following resultwas obtained�

Theorem ���� Let e� � ����� IR be bounded below convex on ���� strictlyconvex at � with e���� � � Then there exists a function W

e� � IR � ���� suchthat

�� We� is increasing�

�� We��� � �

�� We� is continuous at �

�� For all non�negative u � L��)�S� �� �where �)�S� �� is a probability space�with

R�u d� � � the Csisz�ar�Kullback inequality holds�

ku� �kL��d�� � We��e e��u��� �����

where ee��u� ��

R�e��u� d��

For the generating function e���� � � ln� � � �� e�g�� it is known �Csi��� thatWe��c� �

p�c� �����

By a rescaling argument the inequality ����� can be extended to functions u �L��d��� u � � which are not necessarily probability densities� Excluding the caseu � � we replace u by u��R

�u d��� apply ����� and multiply by

R�u d� to get

����u� Z�

u d�

����L��d��

��Z

u d�

�W�

ee��u�

��

where ���� � e��R�ud� ���

Throughout this Section we shall make use of the following assumptions andnotations�

B�� �)�S� �� is a measure space and � is a probability measure on S�i�e� ��)� � �� � � �

B�� � � J � IR is a strictly convex� continuous� non�negative function onthe �possibly unbounded� interval J � which has a non�empty interior J��If � �� inf J �� J � then lim�� ���� � �� If � �� sup J �� J � thenlim�� ���� ���

For functions u � L��d�� �� L��)% d�� we de�ne the entropy

e��u� ��

�����Z�

��u� d�� ���u�� if u�x� � J ��a�e��

� else�

�����

with the notation �u� ��R�u d��

Remark ���� Since � is continuous on J the mapping x �� �� � u��x� is S�measurable whenever u is S�measurable� Furthermore we have due to Jensen�sinequality for all u � L��d�� with u�x� � J �� a�e�Z

��u� d� � ��Z

u d�

��

Thus the right�hand side of ����� has a well�de�ned value in �� ���Remark ���� a� In contrast to De�nition ����� the generating function � doesnot have to satisfy the normalization ���� � ����� � �b� The required continuity of � as well as the assumed behaviour of ���� as �tends to the boundary of J deserve a few more comments� Let -� � J � IR beconvex on the interval J which has a non�empty interior� Then -� is continuouson J� and possible points of discontinuity are contained in �J J� If � � IRand if ���� remains bounded as � approaches � then we can �re��de�ne -���� ��lim���

-����� We get a convex function less or equal to the original one �i�e� theentropy decreases� which is continuous at �� The same procedure applies incases where � is �nite and ���� remains bounded as � approaches �� The cases� � �� or � �� are discussed in c��c� Not every strictly convex continuous function -� � J � IR on the �possiblyunbounded� interval J with non�empty interior satis�es B��� However we cannormalize the generating function -� as in Remark ���� Due to the strict convexityof -� there exist for each �� � J� a constant b � IR such that the function

���� �� -����� -������ b�� � ���

is non�negative on J� Furthermore we have for all u � L��d�� the equalitye��u� � e ���u� which shows that � and -� generate the same entropy�

Remark ���� Let us assume that ) � IRn n � IN is a Borel set S is the Borelsigma algebra on ) and � is a probability measure on S which is absolutelycontinuous with respect to the Lebesgue measure� In this case the Radon�Nikodymderivative of � with respect to the Lebesgue measure exists�

g�x� ��d��x�

dx�

��

To keep things simple let us assume that g�x� for almost all x � )� Nowconsider a function f � L��)� dx� which satis�es

u�x� ��f�x�

g�x�� J a�e� on )�

Then e��u� is � as in the previous sections � the relative entropy of f w�r�t� g�

e��u� � e��f jg� �Z�

��f�g�g dx� ��Z

f dx

��

In contrast to De�nition ��� howeverR�f dx needs not to be normalized here�

In the case of ���� � �ln��� � and f�x� � with R�f dx � � the inequalities

����� and ����� combine to �������

The subsequent analysis aims at generalizing the above entropy estimates� Inparticular we will establish a generalized Csisz�ar�Kullback inequality of the form����u� Z

u d�

����L��d��

� U

�e��u��

Z�

u d�

�� �����

where the function U only depends on � and u belongs to L��d��� We shallpresent a transparent �and almost elementary� construction of U and thus extendthe results of the above cited references to a larger class of functions u �u does nothave to be normalized or non�negative here�� Then we shall prove the optimalityof the estimate and construct U explicitly for certain convex functions �� Theexplicit construction of U is somewhat lengthy and thus deferred to the Appendix�

In the sequel we shall need the following ��dependent constants� For any � � J�we de�ne�

Q����� �� limz��

��� z�� ����z

� ����� if � ���

Q����� �� limz��

��� � z�� ����z

� ����� if � � ���

c���� ��

�������������

����� ���� �� � ��Q����� if �� ���� � IR� � �������� ���� �� � ��Q����� if �� ���� � IR� � � ���

�� � ������ �� � ������� � � � ���� if �� �� ����� ���� � IR�

� else�

��

�with the convention � � �� for all � �� and

Usup��� ��

�������������� � �� if � � IR� � ����� � �� if � � ��� � � IR

��� � ���� � ��� � � if �� � � IR� if � � ��� � ��

We now collect the essential properties of U in the following theorem% its elemen�tary proof is given in the Appendix�

Theorem ���� Under the assumption B�� there exists a function U � ���� �J � IR such that

P�� U is continuous�

P�� For all � � J� U�� �� � �P�� For all � � J� The mapping c �� U�c� �� c � ���� is increasing�P�� For all � � �J J and all c � ���� the identity U�c� �� � holds�

P�� For all � � J�� The mapping c �� U�c� �� c � ���� is strictly increasingon �� c������

P � For all � � J�� limc��

U�c� �� � Usup����

P�� For all � � J� and all c � �c�������� U�c� �� � Usup����

We note that �J J �see P��� may be empty� In P�� we use the convention����� � �� The main result of this Section is�Theorem ��� Assume B�� � B��� Then we have for all u � L��d�� withe��u� �� ����u� Z

u d�

����L��d��

� U

�e��u��

Z�

u d�

�� ��� �

Proof� We �rst shall introduce some abbreviations� Recall that �u� ��R�u d��

Since e��u� � we have u�x� � J ��a�e� Hence �u� � J � We set)�� �� fx � ) � u�x� � �u�g� )� �� fx � ) � u�x� �u�g�

and de�ne

w �� u� �u� � a ��Z���

w d� � � �� ��)�� ��

��

We note that � � �� �� and ��)�� � � � �� SinceR�w d� � � we haveR

��w d� � �a and����u� Z

u d�

����L��d��

Z�

jwj d� �Z���

w d��Z��w d� � �a�

�This factor � also appears in the de�nition of U�� We proceed by a case�distinction� The �rst case is probably the most interesting one�

Case �� � � �� If �u� � �J J � then we will get u � �u� and therefore � � �� Thisis a contradiction since we have assumed � �� Hence �u� � J�� Furthermore weget from the de�nition of a� �u� and ��

a � minf��� � �u��� ��� ����u�� ��g�

and therefore

a � Usup��u��

���� � �u����u�� ��

�� � �

�a

� � �u� � ��a

�u�� �� �� ���

Now we apply Jensen�s inequality�

� e��u�

Z�

���u�� � ��u��� d�

Z�

�� �w �u��� � ��u��� d�

Z���

�� �w �u��� � ��u��� d� Z���� �w �u��� � ��u��� d�

� ���u�

a

�� ���u��

� ��� ��

��

��u�� a

�� ��� ���u��

� a

����u� a

�� ���u��a�

���u�� a

����� ���u��

a���

��� -���u�� a� ��

� inff-���u�� a� �� � � � J��u�� a� �� ��g�

��

where

J��u�� a� ��

�����������������������������

�a

� � �u� � ��a

�u�� ��

if �� � � J�a

� � �u� � ��a

�u�� ��if � �� J� � � J�

a

� � �u� � ��a

�u�� ��

if � � J� � �� J�a

� � �u� � ��a

�u�� ��

if �� � �� J

We have to distinguish between three cases�

Case �a� a Usup��u����� In this case the interval J��u�� a� �� �� has a non�empty interior� Due to the fact that ���� � ��� � �if � � J� and ���� � ������if � � J� we have

e��u� � infn-���u�� a� �� � � � J��u�� a� �� ��

o� inf

�-���u�� a� �� � � �

�a

� � �u� � ��a

�u�� ����

and therefore with the notations of the proof of Theorem ��

I�e��u�� �u�� � a �ku� �u�kL��d��

��

which gives due to the de�nition of U

U�e��u�� �u�� � ku� �u�kL��d���Case �b� a � � This case needs not to be considered here because a � impliesu � �u� and therefore � � �� But this is a contradiction because it is assumedthat � ��

Case �c� a � Usup��u����� Since a � IR we have � � IR or � � IR� If � � IR and� � � we will get the contradiction � � �� � � �� and if � � �� and � � IRwe will get the contradiction � � ��� �� � �� Hence� the only remaining case is�� � � IR� We get � � ��u�� ����� � �� which gives

e��u� � ��u�� ������ �� � �u������� � � � ���u��

� ��u�� ������� �� � �u����� �� � � � ���u��

� c���u���

and therefore by the de�nition of U and Usup��u��

U�e���u��� �u�� � U�c���u��� �u�� � Usup��u�� � �a � ku� �u�kL��d���

Case �� � � �� In this case we easily get u � �u�� Hence� e��u� � � ku ��u�kL��d�� and therefore

U�e��u�� �u�� � U�� �u�� � � ku� �u�kL��d���

We shall now discuss inequality ������

��� Optimality

A natural question in connection with ����� is the following� Is it possible toimprove ������ i�e� is there a function U� � f�c� �� � � � J�� c � �� c������J�g �forreasons explained in Remark ��� below we exclude the cases c � �c������� and� � �J J here� such that

�� for all u � L��d�� with e��u� �� ku� �u�kL��d�� � U��e��u�� �u���

�� there exists a u� � L��d�� with e��u�� � andU��e��u��� �u��� U�e��u

��� �u��� .

Under the assumption B�� �see below� the answer to this question is no�

Theorem ���� Assume B�� B�� and furthermore

B�� For all � � �� �� there exists a set S� � S such that ��S�� � ��

Then for all � � J� and c � �� c����� there exists a sequence �um�m�IN in L��d��such that

�� For all m � IN � �um� � ���� For all m � IN � e��um� � c��� lim

m��kum � �um�kL��d�� � U�c� ���

Proof� Let us consider the case c c���� �rst� In this case there exists a�uniquely determined� a � ��Usup��� such that

c � inf���a���������a�������

-���� a� ���

where -� is de�ned as in the proof of Theorem ���� Clearly� �a � U�c� ��� Wenote that the mapping � �� -���� a� �� is not constant and the mapping a ��inf���a���������a������� -���� a� �� is increasing� Furthermore� -� is continuous� Wecan therefore choose a sequence ��m�m�IN in �� �� and a sequence �am�m�IN in�� a� such that -���� am� �m� � c and limm�� am � a� Now we choose a sequence�Sm�m�IN in S such that ��Sm� � �m for all m � IN � We set for all m � IN

um � ) � IR

x ���

�am��m� � if x � Sm��am���� �m�� � if x � ) n �

We easily verify that �um� � � and e��um� � c� Furthermore�

limm��

kum � �um�kL��d�� � �a � U�c� ���

This �nishes the proof for the case c c����� If c � we choose um � ��which gives �um� � �� e��um� � � c and

kum � �um�kL��d�� � � U�� �� � U�c� ���

Remark ���� In Theorem ��� the cases � � �J J and � � J� with c ��c������� c���� � are not included� Indeed if � � �J J then we wouldhave u � �u�� Hence e��u� � and in this case the estimate ����� is optimaltoo�

ku� �u�kL��d�� � � U�� �� � U�e��u�� �u���

If � � J� and e��u� � c���� we will trivially get for all u � L��d�� with u�x� � J��a�e� and �u� � �

ku� �u�kL��d�� � Usup��u�� � U�c���u��� �u�� � U�c� �u���

We note that this inequality is strict in cases where � � �� or � ���

��� An extension of the inequality

A closer screening of the proof of Theorems �� and ��� shows that the core of allestimates is the fact that for all � � J� and all a � ��Usup��� the strict estimate

inf���a���������a�������

-���� a� ��

holds� For �xed �� � J� this is the case i� � is strictly convex at ��� i�e� i� thereexists a constant b � IR such that the strict estimate

���� ����� b�� � ���

holds for all � � ��� ��� � � ��� We can therefore extend Theorems �� and ���in the following way�

Theorem ���� Assume B�� and

B��� '� � J � IR is a convex continuous non�negative function on the �possiblyunbounded� interval J which has a non�empty interior J�� If � �� J thenlim��

'���� � �� If � �� J then lim��'���� � �� Furthermore assume

that '� is strictly convex at �� � J��

Then�

a� There exists a function U� � ����� IR such that

P�� U� is continuous�

P�� U��� � �

P�� U� is increasing�

P�� U� is strictly increasing on �� c�������

P�� limc��

U��c� � Usup�����

P � For all c � �c��������� U��c� � Usup�����

b� Let u � L��d��� If e��u� � and �u� � �� then����u� Z�

u d�

����L��d��

� U��e ��u���

We note that an optimality result as Theorem ��� also holds for U� �

As an example consider the function '���� � j�j which is strictly convex at �� � �In this case we get ej�j�u� �

R jujd� and U��c� � c which trivially gives for allu � L��d�� with �u� �

kukL��d�� � U��ej�j�u�� � U��

Zjujd�� �

Zjujd��

��� Comparison with the classical Csiszar�Kullback ine�

quality

Inequality ��� deepens the insight into the classical Csisz�ar�Kullback inequalityin four ways� Firstly� inequality ��� is valid for all functions u with e��u� ��i�e� there is no normalizing condition

Rud� � � assumed and u may be negative�

Secondly� we have an optimality result for the function U� Thirdly� it has beenproven that U is strictly increasing on �� c������ Fourthly� the de�nition of Uis rather straightforward and requires no measure�theoretic background but onlyJensen�s inequality�

��� An example

The calculation of U involves a minimizing procedure� If � is di�erentiable thiswill lead to the problem of �nding roots of transcendental equations� Hence thereis no hope for an explicit presentation of U in general�

In some cases� however� the function U can be found explicitly� at least for specialvalues of �u� �

R�u d�� If ���� � j� � �j�p� p � ������ on J � IR� we have for

all admissible u with �u� � ��

ku� �kL��d�� ��Z

�juj�p � �� d�����p

In the situation described in Remark ��� this becomes with the additional requi�rement

R�f dx � ��

kf � gkL��dx� ��Z

jf � gj�p g���p dx����p

For the special case of p � � and �u� � � the above two inequalities read�

ku� �kL��d�� �sZ

�u� � �� d��

and respectively�

kf � gkL��dx� �sZ

�f � g�� g�� dx�

��� Asymptotic behavior of U as e��u�� �

As mentioned before U can in general not be expressed in terms of elementaryfunctions� For �small� values of e��u�� however� one can obtain an asymptoticexpansion of U� Let us recall the following result of �Csi���� If � � C��J�� � � J�and ������ � then the Cszis�ar�Kullback inequality holds�� �K��� � u � L��d��� u � � �u� � � � If e��u� � K� then

ku� �kL��d�� � K�

pe��u��

�����

In �Csi��� the investigations are specialized to the case ���� � � ln�� A strongerresult holds in this case�

For all u � L��d�� with u � R�u d� � ��

ku� �kL��d�� �s�

Z�

u lnu d��

We note that ������ was essential to get ������ Otherwise one can� in general�not get an estimate of the form

ku� �kL��d�� � K �e��u��� � K� � � �����even for small values of e��u� �to keep things simple we consider only the caseR�u d� � ���

Example ����� Let ) � �� �� d� � dx� Let

� � IR� IR � � ��Z

��s� ds

with

� � IR � IR

� ��

�������������� � �� � � � ���� ���e������ � � � �� ��� � � � ��e������ � � � ��� ���e���� � �� � � � ��� ���

Certainly � is bounded below strictly convex and belongs to C��IR� with ���� ������ � ������ � � For k � IN let

uk � �� �� � IR

� ���� ���k� � � � �� ������ ���k� � � � ����� �� �

For all k � IN we have uk � L��dx� uk � R�uk dx � � and kuk � �kL��d�� �

��k� Furthermore for all k � IN �

e��uk� ��

��

�k �

k

� �

�k � �k

���

Hence

limk��

kuk � �kL��d�� � limk��

e��uk� � �

Now let � � ����� Then a simple application of the de l�Hospital rule gives

limk��

kuk � �k���L��d��

e��uk��

��� e�lim��

a�����e�� ���

The core of this example is the fact that all derivatives of � vanish at ��

However� if � is m�times �m � �� continuously di�erentiable in a neighborhoodof �u� � J� with

����u�� � � � � � ��m�����u�� � � ��m���u��

�due to the convexity of �� ��m���u�� has to be non�negative�� we can use theobservation that for all �u� � J�� all su!ciently small a � ���� and � � �a�����u��� �� �a���u�� ���� �see the proofs of Theorems �� and �����

���u� a

�� ���u��a�

���u�� a

����� ���u��

a���

� � ��u� a�� ���u��a

� ��u�� a�� ���u��

a�

�����

Now we can use a Taylor�expansion argument to obtain

if m is even���u� R

�u d�

��L��d��

�� o���� � �

�m+ e��u�

���m��R

�u d�

����m

if m is odd���u� R

�u d�

��L��d��

� o��e��u����m�����

�����

as e��u�� �

The upper estimate can be re�ned in the following cases�

Case �� ���u�� � ����u�� � � �����u�� and � has a logarithmic subentropy�Then � as it is shown in Section � � the term o��� can be ignored and the estimate

ku� �u�kL��d�� �s� e��u�

�����u�������

holds�

Case �� ���u�� � ����u�� � � �����u�� � and ���� � in a neighborhood of�u� �

R�u d�� Then the term o��� is nonnegative and we obtain ����� for all

admissible u with �u� � J� as e��u�� � We note that this estimate holds for alle��u� � ���� whenever � � C��J� with �����u�� and ���� � on J � As anexample consider ���� � e � e �� We have

ku� �kL��d�� �s�

Z�

�eu�� � �� d��

for all admissible u with �u� � ��

Let us keep the assumption that � is three times di�erentiable in a neighborhoodof �u� with �����u�� � Then the estimate ����� is in general not optimal�Consider e�g� ���� � � �ln� � ln�u� � ��� where �u� � ����� We obtain from����� for all admissible u�

ku� �u�kL��d�� �� o���� �s��u�

�Z�

u lnu d�

�� ��u�� ln�u��

where due to ���� � the term o��� is negative� while on the other hand �asmentioned above� we have for all admissible u with �u� � ��

ku� �kL��d�� �s�

Z�

u lnu d��

The reason for this is the fact that the Taylor series argument used to obtain����� provides in general only an upper estimate for U� Whenever this estimate isnot sharp � e�g� for ���� � � ln��� � estimate ����� can be improved� In general�however� this improvement requires a more detailed discussion of �� Let us recallthat in Section � such a detailed discussion was actually carried out� If � admitsa logarithmic�type superentropy� then estimate ����� will hold for all values ofe��u� ��

If the function � is not di�erentiable at �u� �R�u d� � J�� then �����u�� �

�����u�� and we get from ���������u� Z�

u d�

����L��d��

� � e��u�

�����u��� �����u��

as e��u�� �

We can draw the following conclusion from this discussion� The more derivativesof ���u�� vanish �where �u� � J��� the slower u converges to �u� in L��d�� ase��u�� �

�� Passing from weak to strong convergence in L��d��

Inequality ����� allows to pass from weak L��d���convergence to strong L��d���convergence in the following sense� Given a sequence �uk�k�IN in L��d�� and aK � IR with

i�R�uk d�� K as k �� �this is� e�g� the case if uk u as k �� weakly

in L��d��� where u � L��d����ii� There exists a strictly convex and bounded below function � � J � IR �Jis not necessarily open� such that�

ii�a� For all su!ciently large k � IN � uk�x� � J ��a�e��ii�b� lim

k��

Z�

��uk� d� � ��K��

Then

uk � K strongly in L��d�� as k���

The non�trivial fact is that no growth conditions on � are imposed� Considere�g� the case where ���� � e� and let �hk�k�IN be a sequence in L��dx� and leth � L��dx� such that h�x� for almost all x � )� Then by setting d� � h�x� dxwe see that

limk��

Z�

hk dx �

Z�

h dx

together with

limk��

Z�

e�hkh��h h dx � e�h� �h�

implies hk � h strongly in L��dx� as k���

��� The optimal inequality for the logarithmic entropy

Now we consider the case ���� � �ln��� � with J � ���� � ��� ��� ��� � �and �u� � �� We recall that in this case we have for all u � L��d�� with e��u� �the estimates

ku� �kL��d�� � U

�Z�

u lnu d�� �

�� �����

�from ������ and

ku� �kL��d�� �s�

Z�

u lnu d�� ������

�Csisz�ar�Kullback inequality�� In the following plot �Figure �� we compare thefunctions U�c� �� �solid line� and

p� c �dashed line��

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

c

Figure �� U�c� �� �solid� andp� c �dashed� for ���� � �ln� � � �

We observe that for all u � L��d�� with �u� � � and u�x� � �� a�e� the estimate

ku� �kL��d�� � � Usup��� � ���� �

holds� Hence the estimate given by ������ provides no information when e��u� ��� From ������ however� one gets a non�trivial bound of ku��kL��d�� for all �nitevalues of e��u��

��� Entropy�type estimates on kf � gkL��d��

In this subsection we investigate possibilities to estimate kf � gkL��d�� in termsof

e��f jg� �Z�

��f�g�g d��

where we make the assumptions

B�� �)�S� �� is a measure space and � is a probability measure on S�i�e� ��)� � �� � � �

B��$ � � J � IR is a convex� continuous� non�negative function on the �possiblyunbounded� interval J � � � J�� � is strictly convex at �� and ���� �� If � � inf J � J � then lim�� ���� � �� If � � sup J � J � thenlim�� ���� ���

B�� g � L��d�� is positive for ��almost all x � ) and R�g d� � ��

B�� f � L��d�� satis�es f�x��g�x� � J for ��almost all x � )�

If we additionally assume that � is strictly convex� then we obtain from Theorem��

kf � gkL��d�� � U �e��f jg�� ���f ��� �f �� j�f �� �j � ������

with the notation �f � �R�f d��

������ can be applied whenever e��f jg� and �f � are known� The question ariseswhether kf � gkL��d�� can be estimated just in terms of e��f jg�� We will give thesurprisingly simple� a!rmative answer to this question below� To prepare thediscussion we shall introduce some abbreviations �rst� We de�ne

J� �� f� � ���� � � � � Jg � J� �� f� � ���� � �� � � Jg�

and set

�� � J� � ����� �� ����� �� ��� ��

��� � J� � ����

� �� ����� �� ���� �� �

Clearly � are continuous� non�negative� convex functions which are strictly con�vex at � Their respective epigraphs

epi��� � f��� z� � z � ����g

are due to B���� closed and convex subsets of IR�� Following �EkTe��� it is easyto see that the closure of the convex hull of epi���� � epi���� is given by

co �epi���� � epi����� � epi�co���� �����

where the function co���� ��� � J� � J� � ���� is the envelope of all a!nefunctions lower or equal to �� respectively� Some properties of co���� ��� arecollected in the following lemma�

Lemma ����� Assume B���� Then�

a� co���� ��� is convex�

b� co���� ��� is continuous�

c� co���� ��� is strictly increasing�

d� co���� ����� � and co���� ������ for all � � J� � J� � �

The proof of Lemma ���� is deferred to the appendix�

Remark ����� Due to a� and d� of Lemma ���� the function co���� ��� isstrictly convex at � One may ask whether co���� ��� is strictly convex onJ� � J� whenever �� and �� are strictly convex� The following example gi�ves a negative answer to this question� Let �� � ���� � IR� � �� �� and let�� � ����� IR� � �� ��� Then it is easy to see that

co���� ��� � ���� � IR

� ��

����������� � � � �� �

p��

�p�

�� � ��

��� � � ��

p�� �

p��

�� � � � ���

p����

i�e� co���� ��� is not strictly convex�

We set

e�� �� sup�J� J�

co���� �������

The main result of this subsection is

Theorem ����� Assume B�� B��� and B��� Then�

��

a� For all f � L��d�� satisfying B�� the following estimate holds�

kf � gkL��d�� ���co���� ������ �e��f jg�� � e��f jg� e��

maxf�� �� � � �g � e��f jg� � e��

b� If there exists for all � � �� �� a set S� � S such thatRS�g d� � � then

there exists for all � � J� � J� a sequence �fn�n�IN in L��d�� such thatB�� is satis�ed for all n � IN and

kfn � gkL��d�� � � for all n � IN �

� � limn��

�co���� ���

����e��fnjg���

The proof of Theorem ���� is deferred to the appendix� Part b� of Theorem���� is an optimality result �the knowledge of e��f jg� does not allow to concludemore about kf � gkL��d�� than stated in Theorem ����� which applies� e�g� forthe n�dimensional Lebesgue�measure d� � dx� The calculation of co���� ��� isvery simple in cases where �� � �� with J� � J� �or �� � �� with J� � J��holds� In such situations we have

co���� ��� � �� �or co���� ��� � ����

As an example consider ���� � � �ln� � �� �� Then J� � �� ��� J� � ����with

����� � �� �� �ln�� ��� �� � � ��� �� �ln��� ��� �� � � ������for all � � �� ��� Thus co���� ������ � �� �� �ln�� ��� �� � and we obtainProposition ����� Assume B�� B��� and B��� Then for all f � L��d�� withf � the estimate

�� kf � gkL��d����ln�� kf � gkL��d���� �� ��Z�

f ln�f�g� d�

Z�

�f � g� d�

holds�

� Nonlinear Model Problems

We conclude this paper with an application of the theory presented in the previousSections to nonlinear Fokker�Planck type equations�

��

��� Desai�Zwanzig type models

Firstly� consider the following model describing the interaction of a system ofcoupled oscillators in the thermodynamic limit�

�t � Ddivx�rx� rxA�x� �% ��t����� t � � ��a�

��t � � � �I�x� ��� � ��b�

where x � IRn and the parameter vector � � ���� � � � � �M� � IRM � D is a positivedi�usion constant� � presents noise in the oscillator interaction and the oscillatorensemble potential A is given by

A�x� �% ��t�� � ��c�

��

D

�W �x�

�jz��t�� xj� � x �

MXl�m�

s��m�t�Elm�l �

MXl�m�

Elms��l�t� � s��m�t���

Here W is the single oscillator potential �which we assume to be purely determi�nistic� i� e� without noise�� � is a nonnegative parameter �interaction strength�and E � �Elm�l�m����� �M is a symmetric real matrix� on whose largest eigenvaluee� we will impose conditions later� We have

�C�� E � e�I

� in the sense of p�d� matrices�� The nonlinearity in the model stems from theoccurrence of the moments

z��t� ��

ZIRM�

ZIRnx

x��x� �� t�dxdP ���� � ��d�

s��m�t� ��

ZIRM�

ZIRnx

�mx��x� �� t�dxdP ���� � ��e�

The probability measure P represents the distribution of the noise ��

For the following we shall assume

�I � L���IRM� � IRnx% dxdP ���� andZIRM�

ZIRnx

�IdxdP � ��

For more information on the physics of the problem� an extensive list of referencesand a preliminary asymptotic analysis we refer to �ArBoMa� ��

��

We start the analysis by de�ning

���x� �% ��t�� �� exp��A�x� �% ��t��� � ���

and rewrite � ��a� in the usual way

�t � Ddivx

���rx�

���

�� � ���

We set up the relative entropy�type functional

e��j��� ��ZIRM�

ZIRnx

� ln��

���dxdP ���� � ���

Note that� strictly applying De�nition ���� e��j��� is not a relative entropy since�� is not necessarily normalized to � in L

��dxdP ����� Actually� the normalizationof �� was chosen such that

d

dte��j��� � I��j��� � � �

with the entropy production

I��j��� � �DZIRM�

ZIRnx

���jrx�

���j����dx�dP ��� � � ���

�cf� �ArBoMa� ��� We now proceed as in the linear case in Section � and computethe entropy production rate

d

dtI��j��� � ��I��j��� �D�

ZIRM

ZIRn

�MX

l�m�

��zl�xm

��

dxdP � ���

�D�

�ZIRM

ZIRn�jzj�dxdP � j

ZIRM

ZIRn�zdxdP j�

� �D

ZIRM

ZIRn

�z����W

�x�� I

�zdxdP

��DMX

l�m�

Elm

ZIRM

ZIRn

�l�zdxdP �ZIRM

ZIRn�m�zdxdP�

where z �� rx ln������ We remark that this computation follows the lines of the

proof of Lemma ���� where� in addition� the terms which come from the timedependence of ��� have to be taken care of �cf� �ArBoMa� � for details�� Todetermine in � ��� appropriately we assume that W is uniformly convex

�C�� �a � ��W�x��x� � aI x � IRn�

��

Since ZIRM

ZIRn

�l�zdxdP

� � ZIRM

��l dP ���

ZIRM

ZIRn

�jzj�dxdP

we can bound the last term on the right hand side of � ��� �from below� by

��De��ZIRM

ZIRn

�jzj�dxdP

assuming on the second moment of P�

�C��RIRM

j�j�dP ��� � � ��

Altogether we have

d

dtI��j��� � ��I��j��� f�t� � ���

where f�t� � if exists such that

�C�� a� e�� � �

Clearly� this is the case if either e� � �i� e� E is negative semi�de�nite� or� ife� � then the product e�� has to be smaller than the convexity bound a�

Assume now that satis�es �C��� Then � ��� implies

jI��j����t�j � e���tjI��I j���t � ��j� � ���

Since

,z��t� � �DZIRM

ZIRn�zdxdP� ,s��m�t� � �D

ZIRM

ZIRn

�m�zdxdP

we estimate �using I��j��� � �DRIRM

RIRn �jzj�dxdP ��

j ,z��t�j � e���tjI��I j���t � ��j� � ��a�

j ,s��m�t�j � �e���tjI��I j���t � ��j� � ��b�

Now we normalize ���

���x� �% ��t�� �����x� �% ��t��R

IRn���y� �% ��t��dy

ZIRn

�I�y� ��dy � ����

assuming

�C �RIRM e j�j

�dP ��� � for some � su!ciently large�

Obviously we have

I��j��� � I��j����Since

RIRnx��x� �� t�dx is conserved by the equation � ��a� we can apply the loga�

rithmic Sobolev�inequality ����� �with ���� � � ln��� � and a��Das convexity

constant�� ZIRn� ln

��

��

�dx � D

��a ��

ZIRn

�jr� ����j�dx�

After integration with respect to dP ���� �I��j��� appears on the right hand sideand the relative entropy of ��t� with respect to ���t� on the left hand side� Thus

e���t�j���t�� � De���t

��a ��jI��I j���t � ��j � ����

follows and the Csisz�ar�Kullback inequality gives�

k��t�� ���t�kL��dxdP ���� � ��DjI��I j���t � ��j

��a ��

� ��

e��t� � ����

The estimates � ��� imply that z���� �� limt�� z��t� and s��m��� �� limt�� s��m�t�exist and

jz��t�� z����j � �

�jI��I j���t � ��je���t� � ���a�

js��m�t�� s��m���j � ��jI��I j���t � ��je���t� � ���b�

Using these estimates we can �eliminate� the t�dependence of ���x� �% ��t�� asym�ptotically and prove

Theorem ���� Let �C���C�� hold and assume jI��Ij���t � ��j �� Then asteady state '�� � L���dxdP ���� of ����� exists and there is a constant K only depending on �I and on the parameters of the problem ����� such that

k��t�� '��kL��dxdP ���� � Ke��t t �

Clearly� '�� is a solution of the equation

'���x� �� ����x� �% '���R

IRn���y� �% '���dy

ZIRn

�I�y� ��dy� � �� �

��

If W �x� � W ��x� x � IRn holds� then a solution of � �� � can be easily con�structed� We set z� � s��m � and

'���x� �� � N exp�� �D�W �x�

�jxj��

�ZIRn

�I�y� ��dy� � ����

where N ��R

IRn exp�� �

D�W �y� �

�jyj��� dy���� It is immediately veri�ed that

this function solves the equation � �� ��

We remark that convergence of ��t� to a steady state '�� without a convergencerate can still be shown if the single oscillator potential W �x� is not uniformlyconvex but satis�es only certain mild growth conditions �cf� �ArBoMa� ��� Then

it was shown in �ArBoMa� � that I�t�t���� still holds �without a rate�� The

method used above implies immediately

e���t�j��� t����

and ��t�t���� '�� in L��dxdP ���� follows�

��� The drift�di usion�Poisson model

Secondly� we consider the drift�di�usion model with Poisson�coupling for an elec�tron gas �Mar���� �MaRiSc��

�t � div

�r� r

� jxj�� V

��

�� x � IRn� t � � ���a�

��t � � � �I � on IRn�

ZIRn

�I�x�dx � �� � ���b�

�&V � �� � ���c�

Here jxj��acts as a con�ning potential� For the sake of simplicity we only consider

the case of at least � dimensions� i�e� n � � and take the Newtonian solution of� ���c��

V �x� t� ��

�n� ��Sn

ZIRn

��y� t�

jx� yjn��dy� � ���d�

with Sn being the surface area of the unit sphere in IRn� An existence�uniqueness

result �for a global solution� of � ���� is easily obtained by proceeding in analogy

��

to the bounded domain case �cf� e�g� �Gaj� ��� The steady state of � ���� is theunique solution of the mean��eld equation

���x� �� exp��jxj

�� V��x�

��ZIRn

exp

��jyj

�� V��y�

�dy� � ���a�

V��x� ��

�n� ��Sn

ZIRn

���y�jx� yjn��dy� � ���b�

A detailed analysis of equations of the type � ���� can be found in �Dol����

We shall now show that ��t� converges exponentially to �� by using logarithmicSobolev inequalities� We start by de�ning the relative entropy type functional

e ��

ZIRn

� ln

��

��

�dx

ZIRn

jr�V � V��j�dx � ����

�cf� �Gaj� �� and compute its time�derivative�

d

dte�t� � �

ZIRn��t�jr ln

���t�

N�t�

�j�dx � ���

where we denoted the t�local state

N�t� � exp

��jxj

�� V �x� t�

��ZIRn

exp

��jyj

�� V �y� t�

�dy� � ����

We remark that the calculation which leads to � ��� is completely analogous to�Gaj� � making appropriate use of the Poisson equations � ���d�� � ���b��

Assume now that V �t� is in L��IRn� uniformly in t� i�e� there is a K suchthat

kV �t�kL��IRn� � K� t � � ����

�which shall be proven later on under additional assumptions on �I�� Then thelogarithmic Sobolev inequality ����� and the perturbation result of Theorem ���imply the existence of � �independent of t� such thatZ

� ln �N

�dx � �

��

Z�jr ln� �

N�j�dx� t � � � ����

We use � ���� in � ��� and obtain

d

dte�t� � ���

ZIRn

� ln �N

�dx � ���

ZIRn

� ln

��

��

�dx� ��

ZIRn

� ln��N

�dx�

��

Since

ln

���N�t�

�� V �t�� V� ln

�ZIRn

eV��V �t���dx�

we have

d

dte�t� � ���

ZIRn

� ln

��

��

�dx� ��

ZIRn

�V �t�� V���dx ��� ln�

ZIRn

eV��V �t���dx��

�usingRIRn��y� t�dy � � t �� The convexity of f�u� � � lnu implies

� ln�Z

IRn

eV��V �t���dx��ZIRn

�V �t�� V����dx

and

d

dte�t� � ���

ZIRn

� ln

��

��

�dx� ��

ZIRn�V �t�� V�����t�� ���dx

follows� Now we use

��t�� �� � �&�V �t�� V��such that

d

dte�t� � ���

ZIRn

��t� ln

���t�

��

�dx� ��

Zjr�V �t�� V��j�dx

� ���e�t��We conclude exponential convergence in relative entropy�Z

IRn

��t� ln

���t�

��

�dx

ZIRn

jr�V �t�� V��j�dx

� e����t�Z

IRn

�I ln

��I��

�dx

ZIRn

jr�VI � V��j�dx�

� ����

�where� obviously� VI is the Newtonian solution of �&VI � �I� and exponentialL��IRn��convergence of ��t� to �� follows from the Csisz�ar�Kullback inequality�

We are left with proving the bound � ����� Therefore we proceed as in �FaStr�and multiply � ���a� by �q� q � IN� Integration by parts and using � ���c� gives�

q �

ZIRn

�q��dx � � �q

q �

ZIRn

jr�� q��� �j�dx qn

q �

ZIRn

�q��dx� q

q �

ZIRn

�q��dx�

Applying the Nash�inequality �Nas ���ZIRn

f �dx

��� �n

� an�Z

IRn

jf jdx� �

nZIRn

jrf j�dx

��

for some an to the �rst term on the right hand side �with f � �q��� � gives

d

dtzq�� � ��q

an

�zq����� �

n

�z q�����n

nqzq���

where we set

zq ��

ZIRn

�qdx�

It is easy to conclude that zq���t� is uniformly bounded for t if zq���� ��By interpolation we �nd ��t� � Lp��IRn� uniformly for t if �I � Lp��IRn� L���IR

n��

Since the Newtonian potential of a function in Lp�IRn� is in L��IRn� if p n�we

obtain

Theorem ���� Let �I � L���IRn� Lp��IRn� for some p n�� Then there is

a constant � such that the exponential convergence ������ of the relativeentropy holds for the solution ���t�� V �t�� of �������

We refer to �ArMaTo��� for a detailed analysis of bipolar models of the form� ���� �i�e� with two types of carriers��

Appendix Proofs of Section �

Proof of Theorem ����

Step �� We shall �rst construct an auxiliary function �� We introduce

M �� f��� a� �� � J� � ����� �� �� � a minf��� � ��� ��� ���� � ��gg�We note that M � � consists exactly of those ��� a� �� � J� � ����� �� �� forwhich � �a��� and � � �a���� ��� belongs to J�� We de�ne� �M � IR

��� a� �� �� ��� a

�� ����

� ��� ��

��

�� � a

�� ��� ����

��

For �xed � � J� let Pra�M % �� be the projection of M onto the a�axes and for�xed � � J� and �xed a � Pra�M % �� let Pr��M % �� a� be the projection of Monto the ��axis� Then we can easily verify that

Pra�M % �� �

��Usup���

�� Pr��M % �� a� �

�a

� � � � ��a

� � ���

Due to the assumed strict convexity of � we observe that for all ��� a� � J� �� Usup��

��

inf��Pr��M ��a�

���� a� ��

� inf��Pr��M ��a�

a

���� a

�� ����a�

��� � a

����� ����

a���

Step �� We de�ne

-H � J� �� Usup��

�� ����

��� a� �� inf��Pr��M ��a�

���� a� ��

We note that -H is de�ned as the in�mum of a bounded�below� continuous function�recall that each convex function � is continuous on the interior of its domain�over an open set where two entries are �xed� Hence -H is continuous� Furthermore�by the strict convexity of �� the function

-H��� �� �� Usup��

�� ����

a �� -H��� a�

is for all �xed � � J� strictly increasing� It is not very di!cult to check that forall � � J��

lima��

-H��� a� � � lima�Usup���

-H��� a� � c�����

For �xed � � J� we can therefore de�ne the generalized inverse mapping �see�BeLo����

I��� �� � ���� ���Usup���

c ��

�����-H��� ��

����c� if c � �� c�����

Usup���

�if c � �c������

We note that I��� �� is increasing� strictly increasing on �� c����� and that themapping ��� c� �� I�c� ��� � � J�� c � ���� is continuous� Furthermore it iseasy to check that limc�� I�c� �� � � Let us note that

limc�c���

I�c� �� �Usup���

��

��

Step �� Now we are in the position to de�ne U�

U � ����� J � IR

�c� �� ������I�c� �� if c � �� c������ � � J�Usup��� if c � �c�������� � � J� if � � �J J

The veri�cation of P�� �P�� follows from the previous remarks and can thereforebe left to the reader�

Proof of Lemma �����

a�� b� co���� ��� is the envelope of a family of a!ne functions� Hence co���� ���is convex and continuous� see �EkTe����

d� � � and ��� � implies co���� ��� � � We easily deduce from B��$that �������� � ������ for all � � J� � � For each � � J� � J��� � there exists a � � ���� � �� �� such that �� � J� J�� We setK �� minf����������� ����������g� Then it is easy to verify that for alls� � J� and all s� � J��

���s�� � K �s� � ��� � ���s�� � K �s� � ����Hence� by de�nition� co���� ������ � K �� � ��� � ��� �� K � �

c� Let ��� �� � J� � J� with �� ��� We have to prove co���� ������� co���� �������� If �� � � this estimate will follow from d�� If �� � we set � ������������ Certainly � � �� ��� We calculate co���� ������� � co���� ����� �� � �� ��� � �co���� ����� �� � ��co���� ������� co���� �������� sinceco���� ����� � � �� � � �� �� and co���� ������� �

Proof of Theorem �����

a� After the previous remarks it su!ces to prove� For all f � L��d�� satisfyingB��and e��f jg� �� the pair �k�k� e����� �� �kf � gkL��d��� e��f jg�� � co�epi���� �epi������ Since both epi���� and epi���� are convex �subsets of IR�� we have

co�epi���� � epi�����

������ ��� ����� z� � �� � J�� �� � J�� � � �� ���

z � ������� ��� ��������� �Let )�

� �� ff � gg and )� �� ff gg� We de�ne � ��R���g d�� Then � � �� ��

and �� � � R��g d�� We introduce w �� �f�g�� � and set a �� R

���w g d�� as

well as b �� � R��w g d�� Then a� b � ���� and

k�k � a b�

��

We assume for the time being � � � �� Then

e���� � �

Z���

��� w�g

�d� ��� ��

Z����� w�

g

�� � d�

� ����

Z���

�g wg� d�

� ��� ���

��

�� �Z���g wg� d�

�� ��

� a

� ��� ���

��� b

�� ���

Setting

�� ��a

�� �� ��

b

�� ��

we obtain

k�k � ��� ��� ���� � e���� � ������� ��� ��������� ���� �

where �� � J� and �� � J�� The reader may wish to verify ���� � in caseof � � or � � � for some �� � J� and �� � J�� Hence �k�k� e����� �co�epi���� � epi������

b� For all � � J� � J� we have

co���� ������

� inff������� ��� �������� � �� � J�� �� � J�� � � �� ������ ��� ���� � �g�

Hence it is possible to choose a sequence ��n� ��n � �

�n �n�IN in �� ���J��J� with

�n��n ��� �n���n� � � for all n � IN �

limn��

�n�����n � ��� �n������n � � co���� �������

By assumption we can choose for all n � IN a set Sn � S with RSng d� � �n�

Setting for n � INfn � ) � IR

x ����� ��n � g�x� � x � Sn��� ��n � g�x� � x � ) n Sn

�nishes the proof�

��

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