GENERALIZATION OF AN INEQUALITY BY TALAGRAND AND LINKS WITH THE LOGARITHMIC SOBOLEV INEQUALITY
37 pages
English

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GENERALIZATION OF AN INEQUALITY BY TALAGRAND AND LINKS WITH THE LOGARITHMIC SOBOLEV INEQUALITY

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GENERALIZATION OF AN INEQUALITY BY TALAGRAND, AND LINKS WITH THE LOGARITHMIC SOBOLEV INEQUALITY F. OTTO AND C. VILLANI Abstract. We show that transport inequalities, similar to the one derived by Talagrand [30] for the Gaussian measure, are im- plied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately log-concave, in a precise sense. All constants are independent of the dimension, and optimal in certain cases. The proofs are based on partial differential equations, and an interpolation inequality involving the Wasserstein distance, the entropy functional and the Fisher information. Contents 1. Introduction 1 2. Main results 5 3. Heuristics 10 4. Proof of Theorem 1 18 5. Proof of Theorem 3 24 6. An application of Theorem 1 29 7. Linearizations 31 Appendix A. A nonlinear approximation argument 34 References 35 1. Introduction Let M be a smooth complete Riemannian manifold of dimension n, with the geodesic distance (1) d(x, y) = inf ? ? ? √∫ 1 0 |w˙(t)|2 dt, w ? C1((0, 1);M), w(0) = x, w(1) = y ? ? ? . 1

  • dµ d?

  • measure ?

  • d?

  • sobolev inequality

  • gross's logarithmic

  • gotze also

  • d? ?

  • gaussian measure


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GENERALIZATION OF AN INEQUALITY BY
TALAGRAND, AND LINKS WITH THE
LOGARITHMIC SOBOLEV
F. OTTO AND C. VILLANI
Abstract. We show that transport inequalities, similar to the
one derived by Talagrand [30] for the Gaussian measure, are im-
plied by logarithmic Sobolev inequalities. Conversely, Talagrand’s
inequalityimpliesalogarithmicSobolevinequalityifthedensityof
the measure is approximately log-concave, in a precise sense. All
constantsareindependentofthedimension,andoptimalincertain
cases. The proofs are based on partial differential equations, and
an interpolation inequality involving the Wasserstein distance, the
entropy functional and the Fisher information.
Contents
1. Introduction 1
2. Main results 5
3. Heuristics 10
4. Proof of Theorem 1 18
5. Proof of 3 24
6. An application of Theorem 1 29
7. Linearizations 31
Appendix A. A nonlinear approximation argument 34
References 35
1. Introduction
Let M be a smooth complete Riemannian manifold of dimension n,
with the geodesic distance
(1) 8 9s
Z< =1
12d(x;y)=inf jw˙(t)j dt; w2C ((0;1);M); w(0)=x; w(1)=y :
: ;0
12 F. OTTO AND C. VILLANI
We define the Wasserstein distance, or transportation distance with
quadratic cost, between two probability measures „ and ” on M, by
s Z
p
2(2) W(„;”)= T („;”)= inf d(x;y) d…(x;y);2
…2Π(„;”) M£M
where Π(„;”) denotes the set of probability measures on M£M with
marginals „ and ”, i.e. such that for all bounded continuous functions
f and g on M,
Z Z Z
£ ⁄
d…(x;y) f(x)+g(y) = fd„+ gd”:
M£M M M
Equivalently,
n op
2W(„;”)=inf Ed(X;Y) ; law(X)=„; law(Y)=” ;
where the infimum is taken over arbitrary random variables X and Y
on M. This infimum is finite as soon as „ and ” have finite second
moments, which we shall always assume.
The Wasserstein distance has a long history in probability theory
and statistics, as a natural way to measure the distance between two
probability measures in weak sense. As a matter of fact, W metrizes
the weak-* topology on P (M), the set of probability measures on M2
with finite second moments. More precisely, if („ ) is a sequence ofn
probability measures on M such that for some (and thus any) x 2M,0
Z
2lim sup d(x ;x) d„ (x)=0;0 n
R!1 n d(x ;x)‚R0
then W(„ ;„)¡!0 if and only if „ ¡!„ in weak measure sense.n n
Striking applications of the use of this and related metrics were re-
centlyput forwardin worksbyMarton [21] and Talagrand [30]. There,
Talagrandshowshowtoobtainrathersharpconcentrationestimatesin
a Gaussian setting, with a completely elementary method, which runs
as follows. Let
2¡jxj =2e
d (x)= dx
n=2(2…)
denote the standard Gaussian measure. Talagrand proved that for any
nprobability measure „ on R , with density h = d„=d with respect
to ?,
s sZ Z
(3) W(„;?)• 2 hloghd = 2 loghd„:
n n
RRON AN INEQUALITY BY TALAGRAND 3
nNow, let B ‰R be a measurable set with positive measure ?(B),
and for any t>0 let
nB =fx2R ; d(x;B)•tg:t
nHere d(x;B) = inf kx¡ yk . Moreover, let ?j denote the re-y2B B
striction of ? to B, i.e. the measure (1 = (B))d . A straightforwardB
computation, using (3) and the triangle inequality for W, yields the
estimate
s s
¡ ¢ 1 1
W ?j ;?j • 2log + 2log :nB nBt ?(B) 1¡?(B )t
Since, obviously, this distance is bounded below by t, this entails
2
1 1
¡ t¡ 2log
2 ?(B)(4) ?(B )‚1¡e :t
In words, the measure of B goes rapidly to 1 as t grows : this is at
standard result in the theory of the concentration of measure in Gauss
space, which can also be derived from the Gaussian isoperimetry.
Talagrand’s proof of (3) is completely elementary; after establishing
it in dimension 1, he proceeds by induction on the dimension, taking
advantageofthetensorizationpropertiesofboththeGaussianmeasure
and the entropy functional E(hlogh). His proof is robust enough to
yieldacomparableresultinthemoredelicatecaseofatensorproductof
¡ jx jiexponentialmeasure: e dx :::dx ,withacomplicatedvariantof1 n
theWassersteinmetric. BobkovandG otzealsorecoveredinequality(3)
as a consequence of the Pr´ekopa-Leindler inequality, and an argument
due to Maurey [22].
In this paper, we shall give a new proof of inequality (3), and gen-
eralize it to a very wide class of probability measures : namely, all
probability measures ” (on a Riemannian manifold M) satisfying a
logarithmic Sobolev inequality, which means
Z ?Z ¶ ?Z ¶ Z
21 jrhj
(5) hloghd”¡ hd” log hd” • d”;
2‰ hM M M M
holding for all (reasonably smooth) functions h on M, with some fixed
‰>0. Letusrecallthat(5)isobviouslyequivalent,atleastforsmooth
h, to the (maybe) more familiar form
Z ?Z ¶ ?Z ¶ Z
22 2 2 2 2g logg d”¡ g d” log g d” • jrgj d”:
‰M M M M
nIn the case M =R , ” = ?, ‰ = 1, this is Gross’s logarithmic Sobolev
inequality,andweshallprovethatitimpliesTalagrand’sinequality(3).
PRqR4 F. OTTO AND C. VILLANI
As we realized after this study, this implication was conjectured by
Bobkov and G otze in their recent work [5]. But we wish to emphasize
the generality of our result : in fact we shall prove that (5) implies an
inequality similar to (3), only with the coefficient 2 replaced by 2=‰.
Thisresult is in generaloptimal, as showsthe example of the Gaussian
measure. By known results on logarithmic Sobolev inequalities, it also
entails immediately that inequalities similar to (3) hold for (not nec-
¡Ψ(x)¡ˆ(x) nessarily product) measures e dx onR (resp. on a manifold
2M) such that ˆ is bounded and the Hessian D Ψ is uniformly positive
2definite (resp. D Ψ + Ric, where Ric stands for the Ricci curvature
tensor on M).
This implication fits very well in the general picture of applications
of logarithmic Sobolev inequalities to the concentration of measure, as
developed for instance in [19].
Then, a natural question is the converse statement : does an in-
equalitysuchas(3) imply (5)? The answer is known to be positivefor
nmeasuresonR thatarelogconcave,orapproximately: thiswasshown
by Wang, using exponential integrability bounds. But we shall present
acompletelydifferentproof,basedonaninformation-theoreticinterpo-
lation inequality, which is apparently new and whose range of applica-
tions is certainly very broad. It was used by the first author in [26] for
the study of the long-time behaviour of some nonlinear PDE’s. One
interest of this proof is to provide bounds which are dimension-free,
and in fact optimal in certain regimes, thus qualitatively much better
than those already known.
Our arguments are mainly based on partial differential equations.
ThispointofviewwasalreadysuccessfullyusedbyBakryandEmery[3]
to derive simple sufficient conditions for logarithmic Sobolev inequali-
ties (see also the recent exhaustive study by Arnold et al. [1]), and will
appear very powerful here too – in fact, our proofs also imply the main
results in [3].
Note added in proof : After our main results were announced,
S. Bobkov and M. Ledoux gave alternative proofs of Theorem 1 below,
based on an argument involving the Hamilton-Jacobi equation.
Acknowledgement: ThesecondauthorthanksA.Arnold,F.Barthe
and A. Swiech for discussions on related topics, and especially M. Le-
douxforprovidinghislecturenotes[19](whichmotivatedthiswork),as
wellasdiscussingthequestionsaddressedhere. Bothauthorsgratefully
acknowledge stimulating discussions with Y. Brenier and W. Gangbo.
Part of this work was done when the second author was visiting theON AN INEQUALITY BY TALAGRAND 5
University of Santa Barbara, and part of it when he was in the Univer-
sityofPavia; themainresultswerefirstannouncedinNovember,1998,
on the occasion of a seminar in Georgia Tech. It is a pleasure to thank
all of these institutions for their kind hospitality. The first author also
acknowledges support from the National Science Foundation and the
A. P. Sloan Research Foundation.
2. Main results
We shall always deal with probability measures that are absolutely
continuous w.r.t. the standard volume measure dx on the (smooth,
complete)manifoldM,andsometimesidentifythemwiththeirdensity.
¡Ψ(x)We shall fix a “reference” probability measure d” = e dx, and
assume enough smoothness on Ψ : say, Ψ is twice differentiable. As far
nas we know, the most important cases of interest are (a) M =R , (b)
M has finite volume, normalized to unity, and d” =dx (so Ψ=0). An
interesting limit case of (a) is d” = dxj , where B is a closed smooth
B
nsubset of R . Depending on the cases of study, many extensions are
possible by approximation arguments.
Let d„ = fdx, we define its relat

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