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ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
JOSEPH LEHEC
Regularization in L1 for the Ornstein-Uhlenbeck semigroup
Tome XXV, no 1 (2016), p. 191-204.
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Annales de la Faculté des Sciences de Toulouse Vol. XXV, n◦ 1, 2016
pp. 191-204
Regularization in L1 for the Ornstein-Uhlenbeck
semigroup
Joseph Lehec(1)
RÉSUMÉ. — Soit γn la mesure Gaussienne standard sur Rn et soit (Qt) le
semi-groupe d’Ornstein-Uhlenbeck. 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’éliminer le facteur (log log r)4.
ABSTRACT. — Let γn be the standard Gaussian measure on Rn and
let (Qt) be the Ornstein-Uhlenbeck 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.
1. Introduction
Let γn be the standard Gaussian measure on Rn and let (Qt) be the
Ornstein-Uhlenbeck semigroup: for every test function f
Qtf(x) =
∫
Rn
f
(
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 Lehec
Nelson [6] established that if p > 1 and t > 0 then Qt is a contraction from
Lp(γn) to Lq(γn) for some q > p, namely for
q = 1 + e2t(p− 1).
The semigroup (Qt) is said to be hypercontractive. This turns out to be
equivalent to the logarithmic Sobolev inequality (see the classical article by
Gross [3]). In this paper we establish a regularity property of Qtf assuming
only that f is in L1(γn).
Let f be a non-negative function satisfying
∫
Rn
f dγn = 1,
and let t > 0. Since Qtf  0 and
∫
Rn
Qtf dγn =
∫
Rn
f dγn = 1,
Markov inequality gives
γn ({Qtf  r}) 
1
r
,
for all r  1. Now Markov inequality is only sharp for indicator functions
and Qtf cannot be an indicator function, so it may be the case that this
inequality can be improved. More precisely one might conjecture that for
any fixed t > 0 (or at least for t large enough) there exists a function α
satisfying
lim
r→+∞
α(r) = 0
and
γn ({Qtf  r}) 
α(r)
r
, (1.2)
for every r  1 and for every non-negative function f of integral 1. The
function α should be independent of the dimension n, just as the hypercon-
tractivity result stated above. Such a phenomenon was actually conjectured
by Talagrand in [7] in a slightly different context. He conjectured that the
same inequality holds true when γn is replaced by the uniform measure on
the discrete cube {−1, 1}n and the Orstein-Uhlenbeck semigroup is replaced
by the semigroup associated to the random walk on the discrete cube. The
Gaussian version of the conjecture would follow from Talagrand’s discrete
version by the central limit theorem. In this paper we will only focus on the
Gaussian case.
– 192 –
Regularization in L1 for the Ornstein-Uhlenbeck semigroup
In [1], Ball, Barthe, Bednorz, Oleszkiewicz and Wolff showed that in
dimension 1 the inequality (1.2) holds with decay
α(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. They
also 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 Eldan
and Lee who showed in [2] that (1.2) holds with function
α(r) = C
(log log r)4√
log r
,
with a constant C that is independent of the dimension. Again up to the
log log factor the result is optimal.
In this article we revisit the argument of Eldan and Lee. We shall simplify
some steps of their proof and short cut some others. As a result, we are able
to remove the extra log log factor. We would like to make clear though that
this note does not really contain any new idea and that the core of our
argument is all Eldan and Lee’s.
2. Main results
Recall that γn is the standard Gaussian measure and that (Qt) is the
Ornstein-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 > 1
γn ({Qtf > r})  C
max(1, t−1)
r
√
log r
,
where C is a universal constant.
As in Eldan and Lee’s paper, the Ornstein-Uhlenbeck semigroup only
plays a rôle through the following lemma.
Lemma 2.2.— Let f : Rn → R+. For every t > 0, we have
∇2 log(Qtf)  −
1
2t
id,
pointwise.
– 193 –
Joseph Lehec
Proof. — This is really straightforward: observe that (1.1) can be rewritten
as
Qtf(x) = (f ∗ g1−ρ)(
√
ρ x),
where ρ = e−2t and g1−ρ is the density of the Gaussian measure with mean
0 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 appear
anymore.
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 > 1
γn ({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. In
dimension 1, consider the function
fα(x) = e
αx−α2/2.
Observe that fα  0 and that
∫
R fα dγ1 = 1. Note that for every t  1 we
have
γ1 ([t,+∞)) 
c e−t
2/2
t
,
where c is a universal constant. So if α > 0 and r  e then
γ1 ({fα  r}) 
c exp
(
− 12
(
log r
α +
α
2
)2)
log r
α +
α
2
.
Choosing α =
√
2 log r we get
γ1 ({fα  r}) 
c′
r
√
log r
.
– 194 –
Regularization in L1 for the Ornstein-Uhlenbeck semigroup
Since (log fα)
′′ = 0 this shows that the dependence in r in Theorem 2.3
is sharp. Actually this example also shows that the dependence in r in
Theorem 2.1 is sharp. Indeed, it is easily seen that
Qtfα = fαe−t ,
for every α ∈ R and t > 0. This implies that fα always belongs to the image
Qt. Of course, this example also works in higher dimension: just replace fα
by
fu(x) = e
〈u,x〉−|u|2/2
where u belongs to Rn.
Theorem 2.4.— Let X be a random vector having density f with respect
to the Gaussian measure, and assume that f satisfies (2.3). Then for every
r > 1
P (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 let
X be a random vector having density f with respect to γn. Then using
Theorem 2.4
P[f(G) > r] =
+∞∑
k=0
P
(
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 C
max(β, 1)√
log(ekr)
 C e
e− 1
1
r
max(β, 1)√
log r
,
which is the result.
The rest of the note is devoted to the proof of Theorem 2.4.
– 195 –
Joseph Lehec
3. Preliminaries: the stochastic construction
Let µ be a probability measure on Rn having density f with respect to
the 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 argument
shows that we do not lose generality by adding these technical assumptions.
Eldan and Lee’s argument is based on a stochastic construction which
we describe now. Let (Bt) be a standard n-dimensional Brownian motion
and 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 the
Ornstein-Uhlenbeck semigroup. Consider the stochastic differential equa-
tion {
X0 = 0
dXt = dBt +∇ log(P1−tf)(Xt) dt, t ∈ [0, 1]. (3.1)
The technical assumptions made on f insure that the map
x → ∇ logP1−tf(x)
is Lipschitz, with a Lipschitz norm that does not depend on t ∈ [0, 1]. So the
equation (3.1) has a strong solution (Xt). In our previous work [4] we study
the process (Xt) in details and we give some applications to functional
inequalities. 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 by
H(µ1 | µ2) =
∫
log
(
dµ1
dµ2
)
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 values
in Rn which is adapted to the natural filtration of (Bt) (this means that ut
depends only on (Bs)st) and satisfies
∫ 1
0
|ut|2 ds < +∞,
almost surely. Let (vt) be the drift
vt = ∇ logP1−tf(Xt).
Using Itô’s formula it is easily seen that
d logP1−tf(Xt) = 〈vt, dBt〉+
1
2
|vt|2 dt.
– 196 –
Regularization in L1 for the Ornstein-Uhlenbeck semigroup
Therefore, for every t ∈ [0, 1]
P1−tf(Xt) = exp
(∫ t
0
〈vs, dBs〉+
1
2
∫ t
0
|vs|2 ds
)
. (3.2)
Combining this with the Girsanov change of measure theorem one can show
that the random vector X1 has law µ (again we refer to [4] for details).
Moreover we have the equality
H(µ | γn) =
1
2
E
[∫ 1
0
|vs|2 ds
]
. (3.3)
Also, if (ut) is any drift and if ν is the law of
B1 +
∫ 1
0
ut dt,
then
H(ν | γn) 
1
2
E
[∫ 1
0
|us|2 ds
]
. (3.4)
So the drift (vt) is in some sense optimal. Lastly, and this will play a crucial
rôle in the sequel, the process (vt) is a martingale.
Eldan and Lee introduce a perturbed version of the process (Xt), which
we now describe. From now on we fix r > 1 and we let
T = inf{t ∈ [0, 1], P1−tf(Xt) > r} ∧ 1
be 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 T
differs from the one of Eldan and Lee). Now given δ > 0 we let (Xδt ) be the
process defined by
Xδt = Xt + δ
∫ T∧t
0
vs ds.
Since T is a stopping time, this perturbed process is still of the form Brow-
nian motion plus drift:
Xδt = Bt +
∫ t
0
(1 + δ1{sT})vs ds.
So letting µδ be the law of Xδ1 and using (3.4) we get
H(µδ | γn) 
1
2
E
[∫ 1
0
(1 + δ 1{sT})
2|vs|2 ds
]
=
1
2
E
[∫ 1
0
|vs|2 ds
]
+
(
δ +
δ2
2
)
E
[∫ T
0
|vs|2 ds
]
.
(3.5)
– 197 –
Joseph Lehec
4. Proof of the main result
The proof can be decomposed into two steps. Recall that r is fixed from
the beginning and that Xδ1 actually depends on r through the stopping time
T .
The first step is to prove that if δ is small then µ and µδ are not too
different.
Proposition 4.1.— Assuming (2.3), we have
dTV (µ, µ
δ)  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 is
somehow in the direction of ∇f(X1).
Proposition 4.2.— Assuming (2.3), we have
P
(
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 we
need to prove is
P(f(X1) ∈ (r, er])  C
max(β, 1)√
log r
.
We choose
δ =
5
2 log r
.
For this value of δ, Proposition 4.2 gives
P(f(Xδ1 )  e r)  P(f(X1)  r) +
25
4
β + 4
log r
,
whereas Proposition 4.1 yields
P(f(X1)  er)  P(f(Xδ1 )  er) + dTV (µ, µδ)
 P(f(Xδ1 )  er) +
5
2
(
β + 1
log r
)1/2
.
– 198 –
Regularization in L1 for the Ornstein-Uhlenbeck semigroup
Combining the two inequalities we obtain
P(f(X1)  e r)  P(f(X1)  r) + C
max(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 estimate
We actually bound the relative entropy of µδ with respect to µ. Recall
that 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 have
H(µδ | µ)  δ2(β + 1) log r,
for all δ > 0.
This yields Proposition 4.1 by Pinsker’s inequality.
Proof. — Observe that
H(µδ | µ) = H(µδ | γn)−
∫
Rn
log(f) dµδ. (5.7)
Now (5.6) gives
log(f)(Xδ1 )  log f(X1) + 〈∇ log f(X1), Xδ1 −X1〉 −
β
2
|Xδ1 −X1|2,
 log f(X1) + δ
∫ T
0
〈v1, vs〉 ds−
βδ2
2
∫ T
0
|vs|2 ds,
(5.8)
almost surely. We shall use this inequality several times in the sequel. Recall
that X1 has law µ and that X
δ
1 has law µ
δ. Taking expectation in the
previous inequality and using (5.7) we get
H(µδ | µ)  H(µδ | γn)−H(µ | γn)
− δ E
[∫ T
0
〈v1, vs〉 ds
]
+
βδ2
2
E
[∫ T
0
|vs|2 ds
]
.
– 199 –
Joseph Lehec
Together with (3.3) and (3.5) we obtain
H(µδ | µ)  −δ E
[∫ T
0
〈v1 − vs, vs〉 ds
]
+
(1 + β)δ2
2
E
[∫ T
0
|vs|2 ds
]
.
Now since (vt) is a martingale and T a stopping time we have
E
[
〈v1, vs〉1{sT}
]
= E
[
|vs|21{sT}
]
for all time s  1. This shows that the first term in the previous inequality
is 0. To bound the second term, observe that the definition of T and the
equality (3.2) imply that
∫ T
0
〈vs, dBs〉+
1
2
∫ T
0
|vs|2 ds  log r,
almost surely. Since (vt) is a bounded drift, the process (
∫ t
0
〈vs, dBs〉) is a
martingale. Now T is a bounded stopping time, so by the optional stopping
theorem
E
[∫ T
0
〈vs, dBs〉
]
= 0.
Therefore, taking expectation in the previous inequality yields
E
[∫ T
0
|vs|2 ds
]
 2 log r,
which concludes the proof.
6. Proof of Proposition 4.2
The goal is to prove that
P
(
f(Xδ1 )  r1+2δe−4
)
 P(f(X1)  r) + δ2(β + 4) log r.
Obviously
P
(
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 and
rewrite it as
log f(Xδ1 )  K1 + 2δKT + Y
where (Kt) is the process defined by
Kt = log(P1−t)(f)(Xt) =
∫ t
0
〈vs, dBs〉+
1
2
∫ t
0
|vs|2 ds,
– 200 –
Regularization in L1 for the Ornstein-Uhlenbeck semigroup
and Y is the random variable
Y = −2δ
∫ T
0
〈vs, dBs〉+ δ
∫ T
0
〈v1 − vs, vs〉 ds−
βδ2
2
∫ T
0
|vs|2 ds.
Recall that the stopping time T is the first time the process (Kt) exceeds
the value log r if it ever does, and T = 1 otherwise. In particular, if
K1 = log f(X1) > log r
then KT = log r. So if f(X1) > r then
f(Xδ1 ) > r
1+2δ eY .
Therefore
P
(
f(Xδ1 )  r1+2δe−4; f(X1) > r
)
 P(Y  −4).
So we are done if we can prove that
P(Y  −4)  (β + 4)δ2 log r. (6.1)
There are three terms in the definition of Y . The problematic one is
δ
∫ T
0
〈v1 − vs, vs〉 ds.
We know from the previous section that it has expectation 0. A natural way
to get a deviation bound would be to estimate its second moment but it is
not 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. Then
P(Z  −2)  −E[Z].
Remark.— Note that E[Z]  0 by Jensen’s inequality.
Proof. — Simply write
E
[
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 Lehec
Lemma 6.2.— Let Z be the variable
Z = −δ
∫ T
0
〈vs, dBs〉+ δ
∫ T
0
〈v1 − vs, vs〉 ds−
(β + 1)δ2
2
∫ T
0
|vs|2 ds.
Then
P(Z  −2)  δ2(β + 1) log r.
Proof. — As we have seen before the first two terms in the definition of Z
have expectation 0 and
E[Z] = − (β + 1)δ
2
2
E
[∫ T
0
|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 use
the Girsanov change of measure formula. The process (Xδt ) is of the form
Brownian motion plus drift:
Xδt = Xt + δ
∫ T∧t
0
vs ds
= Bt +
∫ t
0
(1 + δ1{sT})vs ds,
Note also that the drift term is bounded. Therefore, Girsanov’s formula
applies, see for instance [5, chapter 6] (beware that the authors oddly use
the letter M to denote expectation). The process (Dδt ) defined by
Dδt = exp
(
−
∫ t
0
(1 + δ 1{sT})〈vs, dBs〉 −
1
2
∫ t
0
∣∣(1 + δ 1{sT})vs
∣∣2 ds
)
is a non-negative martingale of expectation 1 and under the measure Qδ
defined by
dQδ = Dδ1 dP
the process (Xδt ) is a standard Brownian motion. In particular
E[f(Xδ1 )D
δ
1] = 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 exactly
f(Xδ1 )D
δ
1  eZ .
Therefore E[eZ ]  1, which concludes the proof.
– 202 –
Regularization in L1 for the Ornstein-Uhlenbeck semigroup
We now prove inequality (6.1). The idea being that the annoying term
in Y is handled by the previous lemma. Observe that
Y = Z − δ
∫ T
0
〈vs, dBs〉 −
δ2
2
∫ T
0
|vs|2 ds.
So
P(Y  −4)  P(Z  −2) + P
(
δ
∫ T
0
〈vs, dBs〉  1
)
+ P
(
δ2
2
∫ T
0
|vs|2 ds  1
)
.
Recall that
∫ T
0
〈vs, dBs〉 has mean 0 and observe that
E


(
δ
∫ T
0
〈vs, dBs〉
)2
 = δ2E
[∫ T
0
|vs|2 ds
]
 2δ2 log r.
So by Tchebychev inequality
P
(
δ
∫ T
0
〈vs, dBs〉  1
)
 2δ2 log r.
Similarly by Markov inequality
P
(
δ2
2
∫ T
0
|vs|2 ds  1
)
 δ2 log r.
Putting everything together we get (6.1), which concludes the proof.
Acknowledgments
We would like to thank Ronen Eldan and James Lee for introducing us
to the problem and communicating their work to us. We are also grateful to
Michel Ledoux for some fruitful discussions at the final stage of this work.
– 203 –
Joseph Lehec
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[2] Eldan (R.), Lee (J.). — Regularization under diffusion and anti-concentration
of temperature, arXiv:1410.3887.
[3] Gross (L.). — Logarithmic Sobolev inequalities, Amer. J. Math. 97, no. 4,
p. 1061-1083 (1975).
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– 204 –


Regularization in $L_1$ for the Ornstein-Uhlenbeck semigroup

Katsunori  Saito

On the definition of the Galois group of linear differential equations

Federico Lo Bianco

Jorge Vitório Pereira

Smooth foliations on homogeneous compact Kähler manifolds

Joseph Tapia

Alexei Skorobogatov

Alexandre Varchenko

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
JOSEPH TAPIA,
ALEXEI SKOROBOGATOV AND ALEXANDRE VARCHENKO
Preface
Tome XXV, no 2-3 (2016), p. i-i.
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Preface
This volume is a collection of papers in honour of Vadim Schechtman’s
60th anniversary. The diverse fields of mathematics where the papers belong
exemplify Vadik’s broad vision of our beautiful subject. The list of areas to
which he has contributed reads almost like the Mathematics Subjects Clas-
sification: K-theory, algebraic geometry, homological algebra, motivic coho-
mology, representation theory, quantum groups, special functions, mathe-
matical physics,...(have we forgotten anything?). We shall not attempt here
the impossible task to describe in few sentences the finesse and depth of
his influential contributions spanned over more than forty years of math-
ematical research. Yet, to give an idea of the area covered by Vadim, we
will quote for example, special functions, Knizhnik-Zamolodchikov equa-
tions interpreted in terms of a Gauss-Manin connection and root system
of a semi-simple Lie algebra, quantum groups, perverse sheaves and D-
modules, local systems over the complement of hyperplane configurations.
One should also note his beautiful studies on chiral agebras where homolog-
ical and homotopical algebras do interfere together and his studies on the
quantum Langlands correspondence motivated by mathematical physics. It
is however clear to us that this multifaceted work is underpinned not so
much by the chase of new results (of which there is no shortage), but by
a relentless desire to understand mathematics at some deep level. Vadim
adds a taste for the reading of the classics (among his favorites: Gauss’ Dis-
quisitiones Arithmeticae, Jacobi, Eisenstein, Hecke, Weil,...). This readings
are an opportunity for him to regain strength and also to rethink in a new
framework some selected pieces of these great authors.
Perhaps all this could be best described by a joke (told to one of us
by Vadik) which is, as all Jewish jokes are, more a parable than a joke. It
goes like this. A student of a late rabbi is overwhelmed with the wish to see
his former teacher’s reward in the afterlife. After much praying he sees his
master in a vision. But what does he see? The rabbi is sitting in the same
shabby room where he used to sit when he was alive, he wears the same
worn out clothes that he used to wear, and he studies the Torah just like he
always did. Nothing has changed! In dismay the student asks the Holy One
(blessed be He): ”Is this my master’s reward for his righteous and saintly
life, good works and the devout study of the Torah?” ”Yes, was the answer,
but now he understands”.
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Preface

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