Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$ and let
$(Q_t)$ be the Ornstein--Ulhenbeck semigroup. Eldan and Lee recently
established that for every non--negative function $f$ of integral $1$ and any
time $t$ the following tail inequality holds true: \[ \gamma_n ( \{ Q_t f > r
\} ) \leq C_t \, \frac{ (\log \log r)^4 }{r \sqrt{\log r}} , \quad \forall r>1
\] where $C_t$ is a constant depending on $t$ but not on the dimension. The
purpose of the present paper is to simplify parts of their argument and to
remove the $(\log \log r)^4$ factor

Lehec, Joseph

Annales de la faculté des sciences de Toulouse Mathématiques

English

arXiv

Joseph Lehec

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Regularization in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub></mml:math> for the Ornstein-Uhlenbeck semigroup

Annales de la Faculté des Sciences de Toulouse

ANNALESDE LA FACULTÉDES SCIENCESMathématiquesJOSEPH LEHECRegularization in L1 for the Ornstein-Uhlenbeck semigroupTome XXV, no 1 (2016), p. 191-204.<http://afst.cedram.org/item?id=AFST_2016_6_25_1_191_0>© Université Paul Sabatier, Toulouse, 2016, tous droits réservés.L’accès aux articles de la revue « Annales de la faculté des sci-ences de Toulouse Mathématiques » (http://afst.cedram.org/), impliquel’accord avec les conditions générales d’utilisation (http://afst.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelqueforme que ce soit pour tout usage autre que l’utilisation à fin strictementpersonnelle du copiste est constitutive d’une infraction pénale. Toute copieou impression de ce fichier doit contenir la présente mention de copyright.cedramArticle mis en ligne dans le cadre duCentre de diffusion des revues académiques de mathématiqueshttp://www.cedram.org/Annales de la Faculté des Sciences de Toulouse Vol. XXV, n◦ 1, 2016pp. 191-204Regularization in L1 for the Ornstein-UhlenbecksemigroupJoseph Lehec(1)RÉSUMÉ. — Soit γn la mesure Gaussienne standard sur Rn et soit (Qt) lesemi-groupe d’Ornstein-Uhlenbeck. Eldan et Lee ont montré récemmentque pour toute fonction positive f d’intégrale 1 et pour temps t la queuede distribution de Qtf vérifieγn({Qtf > r})  Ct (log log r)4r√log r, ∀r > 1où Ct est une constante dépendant seulement de t et pas de la dimension.L’objet de cet article est de simplifier en partie leur démonstration etd’éliminer le facteur (log log r)4.ABSTRACT. — Let γn be the standard Gaussian measure on Rn andlet (Qt) be the Ornstein-Uhlenbeck semigroup. Eldan and Lee recentlyestablished that for every non-negative function f of integral 1 and anytime t the following tail inequality holds true:γn({Qtf > r})  Ct (log log r)4r√log r, ∀r > 1where Ct is a constant depending on t but not on the dimension. Thepurpose of the present paper is to simplify parts of their argument and toremove the (log log r)4 factor.1. IntroductionLet γn be the standard Gaussian measure on Rn and let (Qt) be theOrnstein-Uhlenbeck semigroup: for every test function fQtf(x) =∫Rnf(e−tx+√1− e−2t y)γn(dy). (1.1)(∗) Reçu le 15/05/2015, accepté le 22/07/2015(1) CEREMADE (UMR CNRS 7534) Université Paris–Dauphine.Article proposé par Fabrice Baudoin.– 191 –Joseph LehecNelson [6] established that if p > 1 and t > 0 then Qt is a contraction fromLp(γn) to Lq(γn) for some q > p, namely forq = 1 + e2t(p− 1).The semigroup (Qt) is said to be hypercontractive. This turns out to beequivalent to the logarithmic Sobolev inequality (see the classical article byGross [3]). In this paper we establish a regularity property of Qtf assumingonly that f is in L1(γn).Let f be a non-negative function satisfying∫Rnf dγn = 1,and let t > 0. Since Qtf  0 and∫RnQtf dγn =∫Rnf dγn = 1,Markov inequality givesγn ({Qtf  r}) 1r,for all r  1. Now Markov inequality is only sharp for indicator functionsand Qtf cannot be an indicator function, so it may be the case that thisinequality can be improved. More precisely one might conjecture that forany fixed t > 0 (or at least for t large enough) there exists a function αsatisfyinglimr→+∞α(r) = 0andγn ({Qtf  r}) α(r)r, (1.2)for every r  1 and for every non-negative function f of integral 1. Thefunction α should be independent of the dimension n, just as the hypercon-tractivity result stated above. Such a phenomenon was actually conjecturedby Talagrand in [7] in a slightly different context. He conjectured that thesame inequality holds true when γn is replaced by the uniform measure onthe discrete cube {−1, 1}n and the Orstein-Uhlenbeck semigroup is replacedby the semigroup associated to the random walk on the discrete cube. TheGaussian version of the conjecture would follow from Talagrand’s discreteversion by the central limit theorem. In this paper we will only focus on theGaussian case.– 192 –Regularization in L1 for the Ornstein-Uhlenbeck semigroupIn [1], Ball, Barthe, Bednorz, Oleszkiewicz and Wolff showed that indimension 1 the inequality (1.2) holds with decayα(r) =C√log r,where the constant C depends on the time parameter t. Moreover the au-thors provide an example showing that the 1/√log r decay is sharp. Theyalso have a result in higher dimension but they loose a factor log log r and,more importantly, their constant C then tends to +∞ (actually exponen-tially fast) with the dimension. The deadlock was broken recently by Eldanand Lee who showed in [2] that (1.2) holds with functionα(r) = C(log log r)4√log r,with a constant C that is independent of the dimension. Again up to thelog log factor the result is optimal.In this article we revisit the argument of Eldan and Lee. We shall simplifysome steps of their proof and short cut some others. As a result, we are ableto remove the extra log log factor. We would like to make clear though thatthis note does not really contain any new idea and that the core of ourargument is all Eldan and Lee’s.2. Main resultsRecall that γn is the standard Gaussian measure and that (Qt) is theOrnstein-Uhlenbeck semigroup, defined by (1.1). Here is our main result.Theorem 2.1. — Let f be a non-negative function on Rn satisfying∫Rn f dγn = 1 and let t > 0. Then for every r > 1γn ({Qtf > r})  Cmax(1, t−1)r√log r,where C is a universal constant.As in Eldan and Lee’s paper, the Ornstein-Uhlenbeck semigroup onlyplays a rôle through the following lemma.Lemma 2.2.— Let f : Rn → R+. For every t > 0, we have∇2 log(Qtf)  −12tid,pointwise.– 193 –Joseph LehecProof. — This is really straightforward: observe that (1.1) can be rewrittenasQtf(x) = (f ∗ g1−ρ)(√ρ x),where ρ = e−2t and g1−ρ is the density of the Gaussian measure with mean0 and covariance (1 − ρ)id. Then differentiate twice and use the Cauchy-Schwarz inequality. Details are left to the reader.What we actually prove is the following, where Qt does not appearanymore.Theorem 2.3.— Let f be a positive function on Rn satisfying∫Rn f dγn =1. Assume that f is smooth and satisfies∇2 log f  −β id (2.3)pointwise, for some β  0. Then for every r > 1γn ({f > r}) C max(β, 1)r√log r,where C is a universal constant.Obviously, Theorem 2.3 and Lemma 2.2 altogether yield Theorem 2.1.Let us comment on the optimality of Theorem 2.1 and Theorem 2.3. Indimension 1, consider the functionfα(x) = eαx−α2/2.Observe that fα  0 and that∫R fα dγ1 = 1. Note that for every t  1 wehaveγ1 ([t,+∞)) c e−t2/2t,where c is a universal constant. So if α > 0 and r  e thenγ1 ({fα  r}) c exp(− 12(log rα +α2)2)log rα +α2.Choosing α =√2 log r we getγ1 ({fα  r}) c′r√log r.– 194 –Regularization in L1 for the Ornstein-Uhlenbeck semigroupSince (log fα)′′ = 0 this shows that the dependence in r in Theorem 2.3is sharp. Actually this example also shows that the dependence in r inTheorem 2.1 is sharp. Indeed, it is easily seen thatQtfα = fαe−t ,for every α ∈ R and t > 0. This implies that fα always belongs to the imageQt. Of course, this example also works in higher dimension: just replace fαbyfu(x) = e〈u,x〉−|u|2/2where u belongs to Rn.Theorem 2.4.— Let X be a random vector having density f with respectto the Gaussian measure, and assume that f satisfies (2.3). Then for everyr > 1P (f(X) ∈ (r, e r])  C max(β, 1)√log r.Theorem 2.4 easily yields Theorem 2.3.Proof of Theorem 2.3. Let G be standard Gaussian vector on Rn and letX be a random vector having density f with respect to γn. Then usingTheorem 2.4P[f(G) > r] =+∞∑k=0P(f(G) ∈ (ekr, ek+1r])+∞∑k=0(ekr)−1E[f(G)1{f(G)∈(ekr,ek+1r]}]=+∞∑k=0(ekr)−1P(f(X) ∈ (ekr, ek+1r])+∞∑k=0(ekr)−1 Cmax(β, 1)√log(ekr) C ee− 11rmax(β, 1)√log r,which is the result.The rest of the note is devoted to the proof of Theorem 2.4.– 195 –Joseph Lehec3. Preliminaries: the stochastic constructionLet µ be a probability measure on Rn having density f with respect tothe Gaussian measure. We shall assume that f is bounded away from 0,that f is C2 and that ∇f and ∇2f are bounded. A simple density argumentshows that we do not lose generality by adding these technical assumptions.Eldan and Lee’s argument is based on a stochastic construction whichwe describe now. Let (Bt) be a standard n-dimensional Brownian motionand let (Pt) be the associated semigroup:Pth(x) = E[h(x+Bt)],for all test functions h. Note that (Pt) is the heat semigroup, not theOrnstein-Uhlenbeck semigroup. Consider the stochastic differential equa-tion {X0 = 0dXt = dBt +∇ log(P1−tf)(Xt) dt, t ∈ [0, 1]. (3.1)The technical assumptions made on f insure that the mapx → ∇ logP1−tf(x)is Lipschitz, with a Lipschitz norm that does not depend on t ∈ [0, 1]. So theequation (3.1) has a strong solution (Xt). In our previous work [4] we studythe process (Xt) in details and we give some applications to functionalinequalities. Let us recap here some of these properties and refer to [4,section 2.5] for proofs. Recall that if µ1, µ2 are two probability measures,the relative entropy of µ1 with respect to µ2 is defined byH(µ1 | µ2) =∫log(dµ1dµ2)dµ1,if µ1 is absolutely continuous with respect to µ2 (and H(µ1 | µ2) = +∞otherwise). Also in the sequel we call drift any process (ut) taking valuesin Rn which is adapted to the natural filtration of (Bt) (this means that utdepends only on (Bs)st) and satisfies∫ 10|ut|2 ds < +∞,almost surely. Let (vt) be the driftvt = ∇ logP1−tf(Xt).Using Itô’s formula it is easily seen thatd logP1−tf(Xt) = 〈vt, dBt〉+12|vt|2 dt.– 196 –Regularization in L1 for the Ornstein-Uhlenbeck semigroupTherefore, for every t ∈ [0, 1]P1−tf(Xt) = exp(∫ t0〈vs, dBs〉+12∫ t0|vs|2 ds). (3.2)Combining this with the Girsanov change of measure theorem one can showthat the random vector X1 has law µ (again we refer to [4] for details).Moreover we have the equalityH(µ | γn) =12E[∫ 10|vs|2 ds]. (3.3)Also, if (ut) is any drift and if ν is the law ofB1 +∫ 10ut dt,thenH(ν | γn) 12E[∫ 10|us|2 ds]. (3.4)So the drift (vt) is in some sense optimal. Lastly, and this will play a crucialrôle in the sequel, the process (vt) is a martingale.Eldan and Lee introduce a perturbed version of the process (Xt), whichwe now describe. From now on we fix r > 1 and we letT = inf{t ∈ [0, 1], P1−tf(Xt) > r} ∧ 1be the first time the process (P1−tf(Xt)) hits the value r, with the conven-tion that T = 1 if it does not ever reach r (note that our definition of Tdiffers from the one of Eldan and Lee). Now given δ > 0 we let (Xδt ) be theprocess defined byXδt = Xt + δ∫ T∧t0vs ds.Since T is a stopping time, this perturbed process is still of the form Brow-nian motion plus drift:Xδt = Bt +∫ t0(1 + δ1{sT})vs ds.So letting µδ be the law of Xδ1 and using (3.4) we getH(µδ | γn) 12E[∫ 10(1 + δ 1{sT})2|vs|2 ds]=12E[∫ 10|vs|2 ds]+(δ +δ22)E[∫ T0|vs|2 ds].(3.5)– 197 –Joseph Lehec4. Proof of the main resultThe proof can be decomposed into two steps. Recall that r is fixed fromthe beginning and that Xδ1 actually depends on r through the stopping timeT .The first step is to prove that if δ is small then µ and µδ are not toodifferent.Proposition 4.1.— Assuming (2.3), we havedTV (µ, µδ)  δ√(β + 1) log r,for every δ > 0, where dTV denotes the total variation distance.The second step is to argue that f(Xδ1 ) tends to be bigger than f(X1).An intuition for this property is that the difference between Xδ1 and X1 issomehow in the direction of ∇f(X1).Proposition 4.2.— Assuming (2.3), we haveP(f(Xδ1 )  r1+2δe−4) P(f(X1)  r) + (β + 4)δ2 log(r),for all δ > 0.Remark.— Note that both propositions use the convexity hypothesis (2.3).It is now very easy to prove Theorem 2.4. Since X1 has law µ, all weneed to prove isP(f(X1) ∈ (r, er])  Cmax(β, 1)√log r.We chooseδ =52 log r.For this value of δ, Proposition 4.2 givesP(f(Xδ1 )  e r)  P(f(X1)  r) +254β + 4log r,whereas Proposition 4.1 yieldsP(f(X1)  er)  P(f(Xδ1 )  er) + dTV (µ, µδ) P(f(Xδ1 )  er) +52(β + 1log r)1/2.– 198 –Regularization in L1 for the Ornstein-Uhlenbeck semigroupCombining the two inequalities we obtainP(f(X1)  e r)  P(f(X1)  r) + Cmax(β, 1)√log r,which is the result.Remark. We actually prove the slightly stronger statement:P(f(X1) ∈ (r, er])  Cmax(max(β, 1)log r,(max(β, 1)log r)1/2).5. Proof of the total variation estimateWe actually bound the relative entropy of µδ with respect to µ. Recallthat log f is assumed to be weakly convex: there exists β  0 such that∇2 log f  −β id, (5.6)pointwise.Proposition 5.1.— Assuming (5.6), we haveH(µδ | µ)  δ2(β + 1) log r,for all δ > 0.This yields Proposition 4.1 by Pinsker’s inequality.Proof. — Observe thatH(µδ | µ) = H(µδ | γn)−∫Rnlog(f) dµδ. (5.7)Now (5.6) giveslog(f)(Xδ1 )  log f(X1) + 〈∇ log f(X1), Xδ1 −X1〉 −β2|Xδ1 −X1|2, log f(X1) + δ∫ T0〈v1, vs〉 ds−βδ22∫ T0|vs|2 ds,(5.8)almost surely. We shall use this inequality several times in the sequel. Recallthat X1 has law µ and that Xδ1 has law µδ. Taking expectation in theprevious inequality and using (5.7) we getH(µδ | µ)  H(µδ | γn)−H(µ | γn)− δ E[∫ T0〈v1, vs〉 ds]+βδ22E[∫ T0|vs|2 ds].– 199 –Joseph LehecTogether with (3.3) and (3.5) we obtainH(µδ | µ)  −δ E[∫ T0〈v1 − vs, vs〉 ds]+(1 + β)δ22E[∫ T0|vs|2 ds].Now since (vt) is a martingale and T a stopping time we haveE[〈v1, vs〉1{sT}]= E[|vs|21{sT}]for all time s  1. This shows that the first term in the previous inequalityis 0. To bound the second term, observe that the definition of T and theequality (3.2) imply that∫ T0〈vs, dBs〉+12∫ T0|vs|2 ds  log r,almost surely. Since (vt) is a bounded drift, the process (∫ t0〈vs, dBs〉) is amartingale. Now T is a bounded stopping time, so by the optional stoppingtheoremE[∫ T0〈vs, dBs〉]= 0.Therefore, taking expectation in the previous inequality yieldsE[∫ T0|vs|2 ds] 2 log r,which concludes the proof.6. Proof of Proposition 4.2The goal is to prove thatP(f(Xδ1 )  r1+2δe−4) P(f(X1)  r) + δ2(β + 4) log r.ObviouslyP(f(Xδ1 )  r1+2δe−4) P(f(X1)  r)+P(f(Xδ1 )  r1+2δe−4; f(X1) > r)Now recall the inequality (5.8) coming for the weak convexity of log f andrewrite it aslog f(Xδ1 )  K1 + 2δKT + Ywhere (Kt) is the process defined byKt = log(P1−t)(f)(Xt) =∫ t0〈vs, dBs〉+12∫ t0|vs|2 ds,– 200 –Regularization in L1 for the Ornstein-Uhlenbeck semigroupand Y is the random variableY = −2δ∫ T0〈vs, dBs〉+ δ∫ T0〈v1 − vs, vs〉 ds−βδ22∫ T0|vs|2 ds.Recall that the stopping time T is the first time the process (Kt) exceedsthe value log r if it ever does, and T = 1 otherwise. In particular, ifK1 = log f(X1) > log rthen KT = log r. So if f(X1) > r thenf(Xδ1 ) > r1+2δ eY .ThereforeP(f(Xδ1 )  r1+2δe−4; f(X1) > r) P(Y  −4).So we are done if we can prove thatP(Y  −4)  (β + 4)δ2 log r. (6.1)There are three terms in the definition of Y . The problematic one isδ∫ T0〈v1 − vs, vs〉 ds.We know from the previous section that it has expectation 0. A natural wayto get a deviation bound would be to estimate its second moment but it isnot clear to us how to do this. Instead we make a complicated detour.Lemma 6.1.— Let Z be an integrable random variable satisfying E[eZ ] 1. ThenP(Z  −2)  −E[Z].Remark.— Note that E[Z]  0 by Jensen’s inequality.Proof. — Simply writeE[eZ] E[eZ 1{Z>−2}] E[(Z + 1)1{Z>−2}]= E[Z]− E[Z 1{Z−2}]+ 1− P(Z  −2) E[Z] + P(Z  −2) + 1.So if E[eZ ]  1 then P(Z  −2)  −E[Z].– 201 –Joseph LehecLemma 6.2.— Let Z be the variableZ = −δ∫ T0〈vs, dBs〉+ δ∫ T0〈v1 − vs, vs〉 ds−(β + 1)δ22∫ T0|vs|2 ds.ThenP(Z  −2)  δ2(β + 1) log r.Proof. — As we have seen before the first two terms in the definition of Zhave expectation 0 andE[Z] = − (β + 1)δ22E[∫ T0|vs|2 ds] −δ2(β + 1) log r.By Lemma 6.1 it is enough to show that E[eZ ]  1. To do so, we usethe Girsanov change of measure formula. The process (Xδt ) is of the formBrownian motion plus drift:Xδt = Xt + δ∫ T∧t0vs ds= Bt +∫ t0(1 + δ1{sT})vs ds,Note also that the drift term is bounded. Therefore, Girsanov’s formulaapplies, see for instance [5, chapter 6] (beware that the authors oddly usethe letter M to denote expectation). The process (Dδt ) defined byDδt = exp(−∫ t0(1 + δ 1{sT})〈vs, dBs〉 −12∫ t0∣∣(1 + δ 1{sT})vs∣∣2 ds)is a non-negative martingale of expectation 1 and under the measure Qδdefined bydQδ = Dδ1 dPthe process (Xδt ) is a standard Brownian motion. In particularE[f(Xδ1 )Dδ1] = E[f(B1)] = 1.Now use inequality (5.8) once again and combine it with equality (3.2)written at time t = 1. This gives exactlyf(Xδ1 )Dδ1  eZ .Therefore E[eZ ]  1, which concludes the proof.– 202 –Regularization in L1 for the Ornstein-Uhlenbeck semigroupWe now prove inequality (6.1). The idea being that the annoying termin Y is handled by the previous lemma. Observe thatY = Z − δ∫ T0〈vs, dBs〉 −δ22∫ T0|vs|2 ds.SoP(Y  −4)  P(Z  −2) + P(δ∫ T0〈vs, dBs〉  1)+ P(δ22∫ T0|vs|2 ds  1).Recall that∫ T0〈vs, dBs〉 has mean 0 and observe thatE(δ∫ T0〈vs, dBs〉)2 = δ2E[∫ T0|vs|2 ds] 2δ2 log r.So by Tchebychev inequalityP(δ∫ T0〈vs, dBs〉  1) 2δ2 log r.Similarly by Markov inequalityP(δ22∫ T0|vs|2 ds  1) δ2 log r.Putting everything together we get (6.1), which concludes the proof.AcknowledgmentsWe would like to thank Ronen Eldan and James Lee for introducing usto the problem and communicating their work to us. We are also grateful toMichel Ledoux for some fruitful discussions at the final stage of this work.– 203 –Joseph LehecBibliography[1] Ball (K.), Barthe (F.), Bednorz (W.), Oleszkiewicz (K.), Wolff (P.).L1-smoothing for the Ornstein-Uhlenbeck semigroup, Mathematika 59, no. 1,p. 160-168 (2013).[2] Eldan (R.), Lee (J.). — Regularization under diffusion and anti-concentrationof temperature, arXiv:1410.3887.[3] Gross (L.). — Logarithmic Sobolev inequalities, Amer. J. Math. 97, no. 4,p. 1061-1083 (1975).[4] Lehec (J.). — Representation formula for the entropy and functional inequalities,Ann. Inst. Henri Poincaré Probab. Stat. 49, no. 3, p. 885-899 (2013).[5] Liptser (R.), Shiryaev (A.). — Statistics of random processes., Vol I, generaltheory. 2nd edition, Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin (2001).[6] Nelson (E.). — The free Markoff field. J. Functional Analysis 12, p. 211-227(1973).[7] Talagrand (M.). — A conjecture on convolution operators, and a non-Dunford-Pettis operator on L1, Israel J. Math. 68, no. 1, p. 82-88 (1989).– 204 –

Regularization in $L_1$ for the Ornstein-Uhlenbeck semigroup

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International audienceLet γn be the standard Gaussian measure on R n and let (Qt) be the Ornstein–Ulhenbeck semigroup. Eldan and Lee recently established that for every non–negative function f of integral 1 and any time t the following tail inequality holds true: γn({Qtf > r}) ≤ Ct (log log r)^4 r √ log r , ∀r > 1 where Ct is a constant depending on t but not on the dimension. The purpose of the present paper is to simplify parts of their argument and to remove the (log log r) 4 factor.Soit γn la mesure Gaussienne standard sur R n et soit (Qt) le semi– groupe d'Ornstein–Ulhenbeck. Eldan et Lee ont montré récemment que pour toute fonction positive f d'intégrale 1 et pour temps t la queue de distribution de Qtf vérifie γn({Qtf > r}) ≤ Ct (log log r)^4/(r√ log r) , ∀r > 1 où Ct est une constante dépendant seulement de t et pas de la dimension. L'objet de cet article est de simplifier en partie leur démonstration et d'´ eliminer le facteur (log log r)^4 

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Regularization in $L_1$ for the Ornstein--Uhlenbeck semigroup

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Regularization in $L_1$ for the Ornstein--Uhlenbeck semigroup