Logarithmic Sobolev Inequalities and Spectral Gaps

Transcription

1 Logarithmic Sobolev Iequalities ad Spectral Gaps Eric Carle ad Michael Loss School of Mathematics, Georgia Tech, Atlata, GA 3033 Jauary 4, 004 Abstract We prove a simple etropy iequality ad apply it to the problem of determiig log Sobolev costats. Itroductio Let µ be a probability measure o of the form dµ = e V (x) dx. Oe says that µ admits a logarithmic Sobolev iequality with costat b i case for all fuctios f o with R f dµ =, f l f dµ b f dµ. (.) There is a extesive literature devoted to the specificatio of coditios uder which there is a logarithmic Sobolev iequality associated to µ, ad the determiatio of the best costat b whe such a iequality does hold. A umber of years ago, it was show by Bakry ad Emery [] that whe V is uiformly strictly covex; i.e., whe the Hessia of V satisfies Hess V (x) ci, (.) for all x, the (.) holds with b = /c. Sice the method of Bakry ad Emery requires that (.) hold uiformly, it does ot apply i may cases of iterest. Oe says that µ admits a spectral gap with costat λ i case for all fuctios u o with R udµ = 0, u dµ u dµ. (.3) λ Takig f to have the form f = α + αu where u is orthogoal to the costats, a simple Taylor expasio yields ( ) f l f dµ = α 3 u dµ + O(α 3 ), (.4) Work partially supported by U.S. Natioal Sciece Foudatio grat DMS c 004 by the authors. This paper may be reproduced, i its etirety, for o-commercial purposes.

2 CL Jauary 4, 004 at least whe u is bouded. Sice i this case we also have, f dµ = α u dµ, it follows that wheever µ admits a logarithmic Sobolev iequality with costat b, it admits a spectral gap with costat λ /b. This useful fact was observed by Rothaus [7]. Ofte it is much easier to prove a spectral gap tha is is to prove a log Sobolev iequality. Ideed, there are may examples of measures µ that admit a spectral gap, but do ot admit a log Sobolev iequality. At this meetig, several problems have bee discussed i which a spectral gap has bee proved, but a log-sobolev iequality has ot or at least ot with useful costats. It ca however be realtively easy to directly establish a restricted log Sobolev iequality: Defiitio I case for some fiite b, (.) is satisifed wheever fdµ = 0 ad f dµ =, (.5) them µ satisifes a restricted log Sobolev iequality. Our aim here is to show that wheever µ has the form µ = e V (x) dx, ad µ admits a spectral gap, the uder broad, easy to check coditios, µ admits a restricted log Sobolev iequality with a explicit costat. From this, we the deduce a urestriced log-sobolev costat for µ. The passage from the restricted to the urestricted iequality based o a simple a priori etropy iequality. We explai this i the secod sectio, ad the i the third sectio, we show how restricted log Sobolev iequalities may be easily established. The restriced log Sobolev iequalities cosidered here are closely related to what are sometimes called deffective log Sobolev iequalities. These take the form f l f dµ b f dµ + a f dµ (.6) for all f with R f dµ =. If µ admits a spectral gap with costat λ, ad (.6) is satisfied, the wheever u satisifies (.5), ( b u dµ + a u dµ b + a ) u dµ λ so that a defective log Sobolev iequality, together with a spectral gap, implies a restricted log Sobolev iequality. There have bee may ivestigatios of log Sobolev iequalities i the settig of diffusio semigroups, startig from the groud breakig work of Gross [6] demostratig the equivalece of hypercotractivity of a diffusio semigroup with the validity of a log Sobolev iequality for the associated Dirichlet form. See [] for a isightful recet survey, ad of particular relevace here, recet work of Cattiaux [3]. I particular, Cattiaux [3] has recetly obtaied log Sobolev iequalities uder coditios similar to those i Theorem. below, however his methods are cosiderably more complicated, ad do ot provide explicit costats.

3 CL Jauary 4, May earlier researchers have also relied o diffusio semigruop argumets. For example, it is well kow that a defective log Sobolev iequality together with a spectral gap implies a log Sobolev iequality. The stadard proof uses Gross s Theorem ad a argumet of Glimm [5]. Gross s Theorem assures that if µ satisfies a defective log Sobolev iequality, the the diffusio semigroup P t associated to the Dirichelt form u dµ is bouded from L to L 4 for some t 0 > 0. Next, the argumet of Glimm is used to show that if a diffusio semigroup P t is bouded from L to L 4 ad some time t 0, ad its geerator has a spectral gap, the for some t > t 0, P t is a cotractio from L to L 4. This cotractivity, together with a iterpolatio argumet ad Gross s Theorem oce agai give the log Sobolev iequality. This sort of argumet is well kow, but rather idirect. Our aim here is to provide a simple, direct passage from defective log Sobolev iequalities to log Sobolev iequalities via spectral gaps, ad also to provide a simple ad direct criterio for the validity of defective log Sobolev iequalities. Because the argumets are simple ad direct, they lead to sharper, more explicit results i may cases. A covexity iequality for etropy Our goal i this sectio is to prove the followig iequality:. THEOREM. Let µ be ay probability measure o a sigma algebra S of subsets of some set. Let u be ay measurable real valued fuctio with u dµ = ad udµ = 0. For all α with 0 α, defie f(x) = α + αu(x). The f l f dµ α + α 4 + α u l u dµ. (.) Before provig the theorem, we make several remarks. First, because u is ormalized i L ad is orthogoal to the costats, f is also ormalized i L. That is, both u dµ ad f dµ are probability measures. Secod, the Taylor expasio (.4) shows that the costat is the best possible costat multiplyig α i (.), for if u(x) = ± for every almost every x, the the right had side of (.) reduces to α + α 4. Third, it might seem atural to try ad prove (.) by cotrollig the remaider terms i the Taylor expasio. However, extractig a estimate o the remaider ivolvig oly α ad u l u dµ does ot seem to be straightforward. Istead, we first prove a L p iequality that is a idetity at p =. Differetiatio i p will yield Theorem.. The L p iequality is the followig:

4 CL Jauary 4, THEOREM. Let µ be ay probability measure o a sigma algebra S of subsets of some set. Let u be ay measurable real valued fuctio with u dµ = ad udµ = 0. For all α with 0 α, defie f(x) = α + αu(x). The for p, f p p ( α ) p/ + p(p ) f (p ) p α u p. (.) Proof: For t real, defie φ(t) by φ(t) = c + tu p dµ where c = α. Differetiatig, we fid φ (t) = p ((c + tu) ) p/ (c + tu)udµ. I particular, φ (0) = p c p c udµ = 0. (.3) Differetiatig oce more, we fid φ (t) = p(p ) c + tu p u dµ. (.4) Applyig Holder s iequality with idices p/(p ) ad p/, we obtai φ (t) p(p ) c + tu p p u p = p(p )φ(t) (p )/p u p. (.5) Together, (.3) ad (.4) show that φ is icreasig i t 0, ad thus, for all t with 0 t α, we have from (.5) that φ (t) p(p )φ(α) (p )/p u p. (.6) The, oce agai usig (.3), φ(α) = φ(0) + Usig the estimate (.6) i (.7) yields the result. α t 0 0 φ (s)dsdt. (.7) Proof of Theorem.: We otice first that (.) holds as a equality at p =. We may therefore differetiate both sides i p at p = to obtai a ew iequality. We first compute [ ( ) d dp f p p = p l f p dµ p ( )] p f p f p l f dµ f p p. p

5 CL Jauary 4, Sice f =, this vaishes at p =. Next, for g = αu, [ d dp g p = p l g p p + ( )] p g p g p l g dµ. p At p =, this reduces to ( ) g g l g dµ = α u l u dµ. Fially, the derivative of p(p )/ at p = is 3/, ad the derivative of c p at p = is c p l c. O the left had side, the derivative of f p p at p = is f l f dµ. Altogether, we have f l f dµ ( α ) l( α ) + 3α + α u l u dµ. (.8) However, by cocavity of the logarithm, so that Usig this i (.8) gives us (.). ( α ) l( α ) ( α )α ( α ) l( α ) + 3α α + α 4. 3 Applicatio to logarithmic Sobolev iequalities I this sectio, we cosider =, ad ad are cocered with the followig questio: dµ = e V (x) d x, Suppose V is such that µ admits a spectral gap with costat λ. What further coditios o V the esure that µ also admits a logarithmic Sobolev costat for some fiite costat b? The followig lemmas provide a positive aswer. The first says that if µ admits a spectral gap, ad if µ admits a restricted log-sobolev iequality, the it admits a urestriced log-sobolev iequality, ad it provides a simple estimate for the costat. 3. LEMMA. Suppose that µ admits a spectral gap λ > 0, ad for some fiite b, u l u dµ b u dµ wheever udµ = 0 ad u dµ =. The µ admits a logarithmic Sobolev iequality with costat o larger tha b + 3 λ.

6 CL Jauary 4, Proof: Cosider ay f with R f dµ =, ad write it i the form cosidered i Theorem.. The, by Theorem., f l f dµ 3α + α u l u dµ. By the spectral gap iequality, α = α u dµ α u dµ f dµ. λ λ By hypothesis α u l u dµ bα u dµ b f dµ. This yields the result. The ext lemma gives coditios uder which a restricted log-sobolev iequality may be prove. 3. LEMMA. Suppose that V is C, ad that µ admits a spectral gap λ > 0, ad { C = if x 4 V (x) } V (x) πe V (x) >. (3.) The for all u satisfyig udµ = 0 ad u dµ =, u l u λ dµ (λ + C )πe u dµ. Before provig Lemma 3., we recall a special case of the family of logarithmic Sobolev iequalities o equipped with Lebesgue measure: For all fuctios g o with R g d x =, g l g d x πe g d x. Proof of Lemma 3.: Cosider ay fuctio u with R udµ = 0, ad R u dµ =. The for ay t with 0 < t <, u dµ tπe u l u dµ R ( ) = ( t) u dµ + t u dµ πe u l u dµ R ( ) ( t)λ u dµ + t u dµ πe u l u dµ. (3.) so that Next, defie V (x)/ g(x) = u(x)e u dµ = g d x.

7 CL Jauary 4, Whe U is smooth with compact support ad V, a simple computatio reveals u dµ = g d x + W g d x where W (x) = 4 V (x) V (x). A stadard approximatio argumet shows idetity, the so called groud state strasformatio is geerally valid. A eve simpler computatio reveals u l u dµ = g l g d x + V g d x. Therefore, u dµ πe u l u dµ R = g d x πe g l g d x ( + W πe V ) g d x R ( W πe V ) g d x. (3.3) Therefore, u dµ tπe u l u [ ( dµ ( t)λ + t W πe V )] g d x. The itegrad is o egative provided that 4 V (x) V (x) πe V (x) + t λ 0 t for all x. Defie C by (3.). The provided C is fiite, we ca chose t = λ/(λ + C ), ad the itegrad will be positive. Lemmas 3. ad 3. immediately yield the followig theorem: 3.3 THEOREM. Suppose that V is C ad that µ admits a spectral gap λ > 0, ad { C = if x 4 V (x) } V (x) πe V (x) >. the µ admits a logarithmic Sobolev iequality with costat b o larger tha λ (λ + C )πe + 3 λ. Notice that if V (x) x γ ad V (x) x γ, the the coditio that C is fiite requires γ. This is cosistet with the fact that wheever µ admits a logarithmic Sobolev iequality with some fiite costat b, there is a umber β > 0 so that dµ <. e β x

8 CL Jauary 4, Thus, cocerig qualitative growth coditios o V, Theorem 3.3 is sharp. It is however surprisig that the Laplacia of V eters C with a egative sig, give that the Bakry Emery coditio implies a logarithmic Sobolev iequality for µ wheever the Hessia of V is uiformly bouded below. To ed the discussio, let us ote that i oe dimesio estimates of the gap are relatively easy to come by. It was show before that with u = e V/ g (.3) reduces to [ V g(x) + 4 V ] g d x λ g d x. ad must hold for all fuctios g satisfyig the coditio R ge V/ d x = 0. I other words, the best possible value for λ is give by the gap of the Schrödiger operator [ V g + V ] g = λg. (3.3) 4 Clearly, the fuctio g 0 = e V/ is the groud state of this Schrödiger equatio with correspodig eigevalue zero. By a elemetary calculatio is easily see that λ is the secod eigevalue of the operator (o L (R, dx)) ( d ) ( ) dx g 0 d g 0 dx g 0. (3.4) g 0 By the well kow commutatio formula (see, e.g. [4] ), the operator ( ) ( d dx g 0 d ) g 0 dx g 0 g 0 has λ as its lowest eigevalue, i fact it has the same spectrum as (3.3) except for the lowest eiegvalue zero. Thus, the gap λ is ow the lowest eigevalue of the operator d dx + V + V 4, which is give by a ucostraied miimizatio. Notice also, that here the secod derivative of the potetial shows up with the right sig. 4 Ackowledgemets This paper grew out of discussios betwe the authors while both were visitig Cedric Villai at E.N.S. Lyo i Jue 003. We thak Cedric for hostig us, ad for may iterestig discussio o a problem of provig a family of log Sobolev iequalities o with costats idepedet of the dimesio. This problem arises i a large deviatios problem cosidered by several authors, ad i particular, Otto ad Villai, ad is explaied i these procedigs. Refereces [] C. Ae et al., Sur les iégalités de Sobolev Logarithmiques, Soc. Math de Frace, Paoramas et Sythèses, No. 0, 00.

9 CL Jauary 4, [] D. Bakry, M. Emery Hypercotractivité de semi groups de diffusio, C. R. Acad. Sci. Paris Sér I Math (984) [3] P. Cattiaux, Hypercotractivity for perturbed diffusio semigroups, preprit, 003. [4] P.A. Deift, Applicatios of a commutatio formula, Duke Math. J (978). [5] J. Glimm, Boso fields with oliear self iteractio i two dimesios, Commu. Math. Phys., 8 5 (968) [6] L. Gross, Logarithmic Sobolev iequalities, Amer. Jour. Math, (976) [7] O.S. Rothaus, Lower bouds for eigevalues of regular Sturm Liouville operators ad the logarithmic Sobolev iequality, Duke Math. Jour (978)