ESAIM: Probability and Statistics
December 1997, Vol. 1, pp. 391{407
OPTIMAL HEAT KERNEL BOUNDS
UNDER LOGARITHMIC SOBOLEV INEQUALITIES
D. BAKRY, D. CONCORDET, M. LEDOUX
Abstract. We establish optimal uniform upper estimates on heat
kernels whose generators satisfy a logarithmic Sobolev inequality (or
entropy-energy inequality) with the optimal constant of R . O-diagonals estimates may also be obtained with however a smaller distance
involving harmonic functions. In the last part, we apply these methods
to study some heat kernel decays for diusion operators on R of the
type ; r rU for some smooth potential U with a given growth at
innity.
n
n
1. Introduction and main result
Let, for example, M be a Riemannian manifold of innite volume, and let dv
denote its Riemannian volume element. In the study of heat kernel bounds,
various functional inequalities have been used in the past years: Sobolev
inequalities Varopoulos (1985)] (with n > 2)
kf k
2
2n=n;2
Z
C jrf j2dv f 2 Cc1(M )
(S)
logarithmic Sobolev inequalities Davies (1989)] or entropy-energy inequalities Bakry (1994)]
Z
Z
Z
n
2
2
2
1
f log f dv 2 log C jrf j dv f 2 Cc (M ) f 2 dv = 1 (LS)
and Nash inequalities Carlen et al. (1987)]
Z
=n C kf k4=n jrf j2 dv f 2 C 1 (M ):
kf k2+4
c
2
1
(N)
As shown in these papers, these three functional inequalities are all equivalent to a common upper bound on the heat kernel pt (x y ) on M given
by
0
(1:1)
sup pt (x y ) C
n=2 t > 0:
xy2M
t
In particular, the inequalities (S), (LS) and (N) are equivalent, for possibly
dierent C > 0 but for the same (analytic) dimension n (> 2 in case of (S)).
URL address of the journal: http://www.emath.fr/ps/
Received by the Journal May 17, 1996. Revised May 15, 1997. Accepted for publication
September 18, 1997.
c Societe de Mathematiques Appliquees et Industrielles. Typeset by TEX.
392
D. BAKRY, D. CONCORDET, M. LEDOUX
This may also be shown directly Bakry et al. (1995)]. Actually, the previous
inequalities belong to a whole family of equivalent inequalities of the type
Z
(1;)=2
2
kf kr C kf ks jrf j dv
f 2 Cc1 (M )
with
1 = + 1 ; 2 0 1
r s
q
and q = 2n=n ; 2. The logarithmic Sobolev inequality (LS) corresponds to
the limiting case r = 2 and = 1 (cf. Bakry et al. (1995)]).
When M = Rn with Lebesgue measure dx, the best constants in the three
inequalities (S), (LS) and (N) are known. Namely
;1
(n + 1)!jB n j 2=n
2
C = 2n(n ; 2) where jB n j denotes the volume of the unit ball B n in Rn, in case of (S)
Aubin (1982)],
(n+2)=n
C = 2((n +N2)=2)
n 2=n
n1 jB j
where N1 denotes the rst non-zero Neumann eigenvalue of the Laplacian
on radial functions on B n , in case of (N) Carlen et al. (1993)]. For (LS), the
task is easier Carlen (1991)]. One may simply start with the logarithmic
Sobolev inequality Gross (1975)] for the canonical Gaussian measure n
on Rn with density 'n (x) = (2 );n=2Rexp(;jxj2 =2), that indicates that, for
every smooth function g on Rn with g 2 dn = 1,
Z
Z
2
2
g log g dn 2 jrg j2dn :
R
Set f 2 = 'n g 2 so that f 2 dx = 1. Then
Z
Z
2
; 2 n=2 jxj2 =2 x
2
dx 2 rf + 2 f dx:
f log f (2 ) e
An integration by parts easily yields
Z
Z
f 2 log f 2 dx 2 jrf j2dx ; n2 log(2) ; n:
f into n=2f (x), > 0, which still satises the normalization
RChanging
2
f dx = 1, shows that, for every > 0 thus,
Z
Z
f 2 log f 2 dx 22 jrf j2 dx ; n2 log(2) ; n ; n log :
R
Optimizing in , we get that for every smooth f on Rn with f 2 dx = 1,
2 Z
Z
n
2
2
2
f log f dx 2 log ne jrf j dx :
(1:2)
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
OPTIMAL HEAT KERNEL BOUNDS
393
Since we started from the logarithmic Sobolev inequality for n with its best
constant for which exponential functions are extremal,
C = n2 e
is the best constant in the inequality (LS) on Rn (with respect to Lebesgue
measure). To further convince ourselves that this constant is optimal, one
may note that among allR functions : R+ ! R+ such that, for every smooth
function f on Rn with f 2 dx = 1,
Z
Z
f 2 log f 2 dx jrf j2dx (1:3)
the function
2u (u) = n2 log n
e
2
n
is actually
R 2 best possible. Indeed, apply (1.3) to f (x) = 'n (x), > 0.
Since f dx = 1, we get
n log 2 ; n n2 :
2
2
2
4
The claim follows by setting u = n2 =4.
Since (S), (LS) and (N) all imply the heat kernel upper bound (1.1), one
may wonder if one of these inequalities with the optimal Euclidean constant
could yield the optimal Euclidean heat kernel bound
sup pt (x y ) 1 n=2 t > 0:
(1:4)
(4t)
xy2M
It is possible that this is the case for the three inequalities. The main result
of this note is that this is the case with the logarithmic Sobolev inequality
(LS). The proof of this result appears as a consequence of the very precise study of uniform upper estimates on the heat kernel under functional
inequalities between entropy and energy developed in Bakry (1994)]. We
present below this result in the context of abstract Markov semigroups of
Bakry (1994)]. We obtain by the same method the sharp bounds on the
norm kPt kpq of the heat semigroup on Rn as an operator from Lp into Lq .
Various comments complement this rst section. In particular, we deduce
from our main result that a complete Riemannian manifold of dimension n
with non-negative Ricci curvature that satises (LS) with the best constant
of Rn is necessarily isometric to Rn. In Section 2, we discuss the o-diagonal
upper bounds on the heat kernel. We namely observe that, again with the
methods of Bakry (1994)], we may obtain the optimal estimates provided
the distance is replaced by a smaller one that takes into account the structure of harmonic functions. In the last section, we use the tool of functional
inequalities between entropy and energy to study diusion operators on Rn
dened by Lf = f ; rf rU for some potential U . According to the
growth of U at innity, we obtain various heat kernel estimates that extend
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
394
D. BAKRY, D. CONCORDET, M. LEDOUX
and clarify some of the results of Kavian et al. (1993)]. After this work was
completed, we became aware of the recent paper Carlen et al. (1995)] by
E. Carlen and M. Loss where the authors obtain optimal estimates for viscously damped conservation laws also relying on Gross's method and on the
sharp logarithmic Sobolev inequality on Rn. They do not consider however
the general implication we establish in Theorem 1.2 below.
We rst turn to (1.4) and describe, to begin with, the framework in which
we will formulate most of our results, refering to Bakry (1994)] for further
details. On some measure space (E RE ), let L be a Markov generator associated to some semigroup Pt f (x) = f (y )pt(x y )d(y ) continuous in L2 ().
We will assume that L and P are invariant and symmetric with respect to
. We assume furthermore that we are given a nice algebra A of (bounded)
functions on E dense in the L2 -domain of L, stable by L and Pt and by the
action of C 1 functions which are zero at zero. (The stability by Pt may
not be satised even in basic examples such as non-degenerate second order
dierential operators with no constant term on a smooth manifold. This
assumption is however not strictly necessary and is mostly used for convenience in order to unify the treatment of a number of various cases that
would require each time a separate standard analysis.) We will mainly be
concerned here with the case when is innite. (Although Bakry (1994)]
is mainly concerned with nite measure spaces, all the results from Bakry
(1994)] used here immediately extend to arbitrary measure spaces.) We denote by kPt kpq , 1 p < q 1, the operator norm of Pt from Lp () into
Lq (). Note that
kPtk11 = sup pt(x y)
where the supremum is understood in the ess sup sense.
We may introduce, following P.-A. Meyer, the so-called \carre du
champ" operator ; as the symmetric bilinear operator on A A dened
by
2;(f g ) = L(fg ) ; f Lg ; g Lf f g 2 A:
Note that ;(f f ) 0. We will say that L is a diusion if for every C 1
function on Rk, and every nite family F = (f1 : : : fk ) in A,
L(F ) = r(F ) LF + rr(F ) ;(F F ):
This hypothesis essentially expresses that L is a second order dierential
operator with no constant term and that we have a chain rule formula for
;, ;((f ) g ) = 0 (f );(f g ), f g 2 A. By the diusion and invariance
properties,
Z
Z
(f )(;Lf )d = 0 (f );(f f )d f 2 A:
One basic operator is the Laplace-Beltrami operator on a complete
connected Riemannian manifold M . In the preceding setting, the \natural" measure is only determined from the Riemannian volume element dv
up to a constant d = cdv . For A the class, say, of Cc1 functions on M
(that is however not stable by the heat semigroup in the non-compact case),
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
OPTIMAL HEAT KERNEL BOUNDS
395
;(f f ) is simply the Riemannian length (squared) jrf j2 of the gradient rf
of f 2 A. The previous abstract framework includes a number of further
examples of interest (cf. Bakry (1994)]). For example, one may consider
L = + X where X is a smooth vector eld on M , or more general second
order dierential operators with no constant term. We may also consider
innite dimensional examples such as the Ornstein-Uhlenbeck generator on
Rn Lf (x) = f (x) ; x rf (x).
In this setting, and following RBakry (1994)], one may consider
general
R
inequalities between the Rentropy f 2 log f 2 d and the energy ;(f f )d of
2
+
+
a function f in A with
R 2 f d = 1. Assume that, for some : R ! R ,
for every f 2 A with f d = 1,
Z
Z
2
2
f log f d ;(f f )d :
(1:5)
In most examples is concave, strictly increasing, and of class C 1 , which
we assume throughout the argument below. Therefore, for every v > 0,
(u) (v ) + 0 (u)(u ; v ), so that (1.5) reads, for every v > 0 and every
f (and by homogeneity),
Z
Z
Z
2
2
0
f log f d (v) ;(f f )d + (v ) f 2 d
where (v ) = (v ) ; v 0 (v ). We thus deal equivalently with a family of
logarithmic Sobolev inequalities. Now, by the diusion property, changing
f > 0 into f p=2 shows that, for every f > 0, v > 0, p > 1,
Z
Z
Z
p
p
p
f log f d ; f d log f p d
Z
Z
2
p
0
p
;
1
; (v) 4(p ; 1) f Lfd + (v) f pd:
Choose now in this inequality a function v ! v (p) > 0, p > 1. According to
Bakry (1994)], we make then use of the fundamental argument of L. Gross
Gross (1975)]. Namely, if
V (t) = e;m(t) kPt f kp(t) t 0
the preceding inequality will show that V 0 0 and V is non-increasing as
soon as p and m are chosen so that
0
p2 (t) = 0 ;v ;p(t) p2 (t)
0 (t) = ;v ;p(t) p (t) :
and
m
p0 (t)
4(p(t) ; 1)
p(t)2
If this is the case, for every 1 p < q 1,
kPtkpq em
(1:6a)
where
Z q ; ds
Z q ; ds
0
t = tpq = v (s) 4(s ; 1) and m = mpq = v (s) s2 (1:6b)
p
p
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
396
D. BAKRY, D. CONCORDET, M. LEDOUX
provided we can nd a function v for which these two integrals are nite. The
optimal choice, that will be used throughout this work (cf. Bakry (1994)]),
is given by v (s) = s2 =(s ; 1), where > 0 is a parameter.
It might be worthwhile noting that, conversely, the previous bounds on
kPtkpq imply that the corresponding entropy-energy inequality (1.5) holds.
This is a consequence of the following proposition.
Proposition 1.1. Under the previous notation, assume that, for some 1 p < 1 and every q in some neighborhood of p, kPtkpq em where t and
m are dened with (1.6b) through some function . Then, for every nonnegative f in A,
Z
Z
Z
p
p
p
f log f d ; f d log f p d
Z
Z
2
p
0
p
;
1
; (v) 4(p ; 1) f Lfd + (v) f p d:
The proof reduces to check that if, for f > 0 in A,
U (") = e;m(") kPt(") f kp+"
where t(") = tpp+" , m(") = mpp+" , then U 0 (0) 0 amounts to the inequality of the proposition.
The main observation of this work is that the general method leading to
the bound (1.6) yields the optimal heat kernel bound (1.4) if we start from
a (LS) inequality for the generator L (or rather the carre du champ ;) with
the best constant of Rn.
R
Theorem 1.2. Assume that for every f in A with f 2 d = 1,
Z
Z
2
n
2
2
(1:7)
f log f d 2 log ne ;(f f )d :
Then,
1 t>0
(4t)n=2
(where the supremum is understood in the ess sup sense).
If
Z
Z
Z
n
2
2
f log f d 2 log C ;(f f )d f 2 d = 1
for some C > 0, by a simple change of variables,
neC n=2
sup pt (x y ) 8t
t > 0:
sup pt (x y ) It is worthwhile mentioning that the logarithmic Sobolev inequality (1.7)
of Theorem 1.2 behaves correctly under tensor product. Namely, if two carres du champ ;1 and ;2 satises such an inequality with dimensions n1 and
n2 respectively, then the product operator ;1 ;2 will satisfy this inequality
with dimension n1 + n2 . This immediately follows from the classic product
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
OPTIMAL HEAT KERNEL BOUNDS
397
property of entropy together with the linearized version of (1.7) (the inequality leading to (1.2)). This stability property is re ected similarly on
the heat kernel bound as can be seen from the example of the Euclidean
spaces.
Proof. We simply use (1.6) with
(u) = n log 2u :
2
ne
Hence 0 (u) = n=2u and
(u) = n2 log n2ue2 :
Then kPt k11 em with
Z 1 ds n
n
t = t() = 8
s2 = 8
1
and
Z 1 2 s2 ds
n
log n e2 s ; 1 s2
m = m() = 2
1
where > 0. From the rst equality, = n=8t. The second yields
Z 1 s2 ds
2
n
n
log s ; 1 s2
m = 2 log n e2 + 2
1
Z1 ;
n
2
n
= log
;
log
r
(1
;
r
)
dr
2
2
n2e 2 0
= n2 log n e2 + n
with the change of variable r = 1=s. Since = n=8t,
m = m() = n2 log 41t
which yields
kPtk11 em = (4t1)n=2
and the result. The proof is complete.
tu
It is clear that the same proof yields upper bounds for kPt kpq for every
t > 0 and 1 p < q 1. Namely kPtkpq em where
Z q ds n 1 1 n
t = tpq () = 8 s2 = 8 p ; q
p
and
Z q 2 s2 ds
n
m = mpq () = 2 log ne2 s ; 1 s2 :
p
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
398
D. BAKRY, D. CONCORDET, M. LEDOUX
After some calculations very similar to the previous ones, we nd that
" 1 ; 1 1; 1 # 2 " # 2 ( 1 ; 1 )
p 1; p
1
1
1
kPtf kpq 1 ; 1 1; 1
:
(1:8)
4t p ; q
q 1; q
p
p
q
q
n
n
p
q
It is less clear however why these bounds should be optimal on Rn. The
next proposition answers this question positively. These bounds (1.8) and
their optimality may also be shown to follow from the work of E. Lieb Lieb
(1990)] on Gaussian maximizers of Gaussian kernels.
Proposition 1.3. With the preceding notation, let be such that (1.5)
holds and kPt k11 = em where
Z 1 s2 ds
t=
0
s ; 1 4(s ; 1)
1
and
Z 1 s2 ds
m=
s ; 1 s2
1
for some > 0. Then, for every 1 p < q 1,
kPt kpq = em
pq
pq
where
Z q s2 ds
tpq = 0 s ; 1 4(s ; 1)
p
and
Z q s2 ds
mpq = s ; 1 s2 :
p
The proof of the proposition is easy. By the hypothesis,
1 k11 = kPt k11 = em = em1 +m
But now also, by the semigroup property,
kPt1
p
+tpq +tq
kPt1
p
p
pq
1:
+mq
1 k11 kPt1 k1q kPt kpq kPt 1 kq1
em1 +m +m 1
+tpq +tq
q
p
pq
pq
q
q
so that it is impossible that kPt kpq < em for some p < q . This result clearly applies in the Euclidean case to prove that (1.8) are actually
equalities in this case. It moreover implies the somewhat surprising following observation. For every 1 p < q < r 1, and every t 0, there is an
unique s such that
kPtkpr = kPskpq kPt;skqr :
Indeed, since 0 (u) = n=2u in this case, one may dene > 0 by
Z r s2 ds
Z r ds
n
0
t = tpr () = s ; 1 4(s ; 1) = 8 s2 :
p
p
pq
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
pq
OPTIMAL HEAT KERNEL BOUNDS
399
Set then s = tqr () and the claim follows from the preceding argument.
In order to e!ciently use Theorem 1.2, it would be worthwhile to know
how to establish (LS) with sharp constant in a Riemannian or abstract setting using curvature-dimension hypotheses. Before inspecting this question
more closely, let us observe the following.
Proposition 1.4. Let M be a Riemannian manifold of dimension n and
non-negative Ricci Zcurvature. If, for every non-negative f in Cc1 (M ),
lim (4t)n=2 Pt f = c fdv, then the heat kernel pt (x y ) on M satises
t!1
sup pt (x y ) xy2M
c
(4t)n=2
t > 0:
Proof. It is an immediate consequence of the Li-Yau inequality Li et al.
(1986)] for Riemannian manifolds with non-negative Ricci curvature, that
ensures in particular that, if f > 0,
; PPftf 2nt
t
for every t > 0 (where Pt denotes the heat semigroup on M ). In another
words, at every point, theR function tn=2 Pt f is increasing in t. But then,
(4t)n=2Pt f increases to c fd so that
kPtk11 (4tc)n=2 :
The proof is complete. Note that by the results of P. Li Li (1986)], M is
isometric to Rn if c = 1 (cf. the proof of Corollary 1.6). Observe furthermore
that the proposition similarly applies to generators ;r log for which the
results of Li et al. (1986)] are also available (in which case the hypothesis on
non-negative Ricci curvature has to be replaced by the condition ;2 (f f ) 1
2
tu
n (Lf ) as explained below).
Z
n=2
What we actually conjecture, is that if tlim
!1(4t) Pt f = c fdv for every smooth f on a manifold M of dimension n and non-negative Ricci curvature, then the logarithmic Sobolev inequality (LS) holds with its sharp
constant from Rn, i.e.
2 c2=n Z
Z
Z
n
2
2
2
f log f dv 2 log ne jrf j dv f 2 dv = 1:
(Proposition 1.4 would then be a consequence of Theorem 1.2.) We have
not been able to prove such a result although we strongly conjecture that
it must be true. So far, we have only been able to prove the inequality
with a constant that misses the optimal one by a factor 1= log 2. We present
this result in the context of the abstract geometry of Markov generators.
Introduce the \iterated carre du champ operator" by setting, for every f g
in A,
2;2 (f g ) = L;(f g ) ; ;(f Lg ) ; ;(g Lf ):
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
400
D. BAKRY, D. CONCORDET, M. LEDOUX
We say that L is of curvature R 2 R and dimension n 1 if for every f in
A,
;2 (f f ) R;(f f ) + n1 (Lf )2 :
By Bochner's formula, the Laplace-Beltrami operator on a Riemannian manifold of dimension n and Ricci curvature bounded below by R is of curvature
R and dimension n in the preceding functional sense.
Proposition 1.5. Let L be a diusion generatorR of curvature 0 and dimension n 1. Assume that, for every f in A with fd = 1,
Z
;
(1:9)
lim sup Pt f log (4t)n=2 Pt f d ; n2
t!1
R
for some > 0. Then, for every f 2 A with f 2 d = 1,
Z
Z
n
2
2
2
f log f d 2 log ne ;(f f )d :
In particular, if (1.9) holds with = 1, the inequality holds with the optimal
constant of Rn.
Proof. It is a simple application of the
R argument developed in Section 6 of
Bakry (1994)]. Let f > 0 in A with fd = 1. By the semigroup properties,
for every t 0,
Z
Z
Zt
f log fd = Pt f log Ptfd + F (s)ds
(1:10)
where
0
Z ;(P f P f )
s s d:
Ps f
Now, in the proof of Proposition 6.7 of Bakry (1994)], it is shown that,
under the curvature-dimension assumption on L,
F 0 (s) n2 F (s)2 s 0:
Hence,
Zt
F (s)ds n2 log 1 + 2nt F (0) :
0
Together with (1.10) and the assumption on the behavior at innity of Pt ,
the conclusion immediately follows (changing f into f 2 ). The proof is complete.
tu
What we expect is that actually (1.9)
with
=
1
appears
as
a
consequence
Z
n=
2
of the fact that tlim
!1(4t) Pt f = fd for every f in A. We only checked
so farR (1.9) with = log 2. Let f be non-negative (for simplicity) and such
that fd = 1. It is easy to see, by Jensen's inequality, that, for every t 0,
Z
Z
;
; ;
Pt f log (4t)n=2 Pt f d = fPt log (4t)n=2 Pt f d
Z
;
f log (4t)n=2P2t f d:
F (s) =
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OPTIMAL HEAT KERNEL BOUNDS
401
In a Riemannian (or concrete) setting, we know (cf. the proof of Proposition
1.4) that (4t)n=2 Pt f is increasing (to 1) so that in this case
Z
;
lim sup Pt f log (4t)n=2 Pt f d ; n2 log 2:
t!1
But Proposition 1.5 then only yields
1 Z
Z
Z
n
2
2
f log f d 2 log n ;(f f )d f 2 d = 1:
n
Note nally that (1.9) with = 1 is satised
R in R . Let f be non-negative
n
on R with compact support and such that fdx = 1. Then, for every x in
the support of f and every t large enough,
Z
Z
2
(4t)n=2 Pt f (x) = e;jx;yj =4t f (y )dy 1 ; 1 jx ; y j2 f (y )dy:
4t
Hence,
;
log (4t)n=2Pt f (x) ; 1
Therefore,
Z
; ;
1
n=
2
Pt log (4t) Ptf (x) ; 4t
Z
= ; 41t
4t
Z
jx ; yj2f (y)dy:
e;jx;y;zj =4tjz j2 f (y )dy dzn=2
(4t)
;
jx ; yj2 + 2tn f (y)dy:
2
(1.9) (with = 1) follows as t tends to innity.
We conclude this section with a comparison theorem for Riemannian manifolds with non-negative Ricci curvature.
Corollary 1.6. Let M be a complete Riemannian manifold of dimension
n with non-negative Ricci curvature satisfying the logarithmic Sobolev inequality (LS) with the best constant of Rn, i.e.
2 Z
Z
Z
n
2
2
2
f 2 dv = 1:
f log f dv 2 log ne jrf j dv Then M is isometric to Rn.
Proof. Denote by V (x r) the volume of the ball with center x 2 M and
radius r > 0 in M . By Theorem 1.2, for every x y 2 M , t > 0,
pt (x y ) (4t1)n=2 :
(1:11)
In particular, by the results of Li et al. (1986)], M has maximal volume
growth, that is
lim
inf V (x r) > 0:
r!1 rn
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402
D. BAKRY, D. CONCORDET, M. LEDOUX
Li's result Li (1986)] then indicates that, for every x y z 2 M ,
; p
jBn j
lim
V
z
t
p
(
x
y
)
=
t
t!1
(4 )n=2
where we recall that jB n j is the volume of the Euclidean unit ball. Together
with (1.11), we thus see that
lim inf V (x r) 1:
(1:12)
r!1
jBn jrn
Now, by Gromov's comparison theorem (cf. Carlen (1991)]), for s < r,
V (x r) V (x s)
jBn jrn jBn jsn
and, in particular, V (x r) jB n jrn as s ! 0. But now, by (1.12), we also
get V (x s) jB n jsn as r ! 1. Therefore V (x r) = jB n jrn for every x 2 M
and r > 0. By the case of equality in the volume comparison theorem, M is
isometric to Rn. The corollary is established.
tu
2. Off-diagonal estimates
As is well-known Bakry (1994)], Davies (1989)], an entropy-energy (or logarithmic Sobolev) inequality of the type (1.5) also leads to o-diagonal upper bounds on the heat kernel in terms of the distance function dened as
d(x y ) = sup f (x) ; f (y )] (the supremum being understood in the ess
;(ff )1
sup sense).
Let us recall the general procedure of the method due to E. B. Davies (cf.
Davies (1989)]). The idea is to derive from an entropy-energy inequality for
the generator L
Z
Z
f log f d 2
2
Z
;(f f )d f 2 d = 1
an entropy-energy inequality for Lh (f ) = e;h L(eh f ), where ;(h h) ,
which depends only on and > 0. This leads, via the method presented in Section 1, to a uniform upper bound on the kernel pht (x y ) =
pt(x y )eh(x);h(y) of the semigroup Pth = e;h Pt(eh ) with generator Lh . We
thus bound in this way pt (x y ) by
C (t )eh(x);h(y) :
Applying this result to h, ;(h h) 1, and optimising in and h, leads to
some bound
pt(x y ) eV (td(xy)) :
Theorem 5.3 of Bakry (1994)] yields an explicit form of the function V in
terms of the entropy-energy function of L. Precisely,
;
pt(x y) eV (td(xy)) for every t T d(x y ) ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
OPTIMAL HEAT KERNEL BOUNDS
403
where the functions V and T are parametrized by and in the following
way: if is dened by
Z 1 0 (s)
p ds
d(x y ) =
then
Z 1 0 (s)
;
p
ds
T d(x y ) = 12
s(s ; )
and if is dened by
then
s
Z1
0
1
p (s)
t= 2
ds
s(s + ; )
r
Z1
;
p 3 (s)
V t d(x y) = ( ) ; 2
ds
s (s + ; )
(where we recall that (s) = (s) ; s0 (s)).
;
Let us use these bounds with the optimal entropy-energy function on Rn
2u (u) = n2 log n
e u > 0:
After some computations, we get that
n=2 n 2
e
d
(
x
y
)
d
(
x
y
)
pt(x y) 8n
exp ; 4t :
t
Unfortunately, this does not give the optimal bound on Rn. In order to
improve this estimate, we introduce a new distance called the \harmonic"
distance dened by
dH (x y ) =
sup f (x) ; f (y )]:
;(ff )1 Lf =0
Note that dH = d on Rn (while dH = 0 on a compact Riemannian manifold).
Proposition 2.1. Assume we have an entropy-energy inequality (1.5). Then
H (x y )2 d
pt(x y ) exp m ; 4t
where t and m are dened in (1.6b) (assumed to be nite).
Proof. The proof follows the same lines as the proof of (1.6a). Fix h with
Lh = 0 and ;(h h) 1 and consider Pth = e;h Pt (eh ) whose generator
is given by Lh (f ) = e;h L(eh f ), > 0. Then, for f 0,
Z
Z ;
p
;
1 h
; f L fd = ; e;h f p;1 ehf d
Z
Z
= ;(f p;1 f )d + (p ; 2) f p;1 ;(f h)d
Z
; 2 f p;(h h)d:
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
404
D. BAKRY, D. CONCORDET, M. LEDOUX
The second term on the right-hand-side is equal to
Z
Z
p
;
2
p
;
2
p
;(f h)d = ;
f p Lhd = 0
so that
p
;
Z
p
f p;1Lh fd ;
Z
f p;1Lfd ; 2
Z
f pd:
Therefore, if, for every v > 0,
Z
Z
Z
f p log f pd ; f p d log f p d
Z
Z
2
p
0
p
;
1
; (v) 4(p ; 1) f Lfd + (v) f pd
then
Z
Z
Z
p
p
p
f log f d ; f d log f p d
Z
2
; 0 (v) 4(pp; 1) f p;1 Lhfd
Z
2
p
2 0
+ (v ) + (v ) 4(p ; 1) f p d:
According to the sketch of proof in Section 1, this leads to the upper bound
kPth k11 em+2 t
where t and m are given by (1.6b). Equivalently,
pt (x y ) em+2 t;h(x);h(y)] :
Optimizing in > 0, and then in h, leads to the claim. Proposition 2.1 is
thus established.
tu
As a consequence of this proposition, if
2u (u) = n2 log n
e u > 0
we get that
dH (x y )2 1
pt(x y ) (4t)n=2 exp ; 4t
for every t > 0 and x y , a formula that is of course optimal on Rn. It would
be worthwhile to study further examples where dH = d, as well as the
various bounds which may similarly be deduced through distances involving
harmonic functions.
Note also that if we are looking for a situation such that (1.5) holds
together with kPt k11 = em in (1.6), then
H
2
pt(x y ) pt (x x) exp ; d (4xt y)
for every t > 0 and x such that pt (x x) = supu pt (u u).
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
OPTIMAL HEAT KERNEL BOUNDS
405
3. Explicit bounds for some Laplacian
+ potential operators in Euclidean space
In this last section, we investigate heat kernel bounds for operators on Rn
given by Lf = f ; rU rf where U is some smooth function whose
growth at innity will determine upper estimates on the heat kernel for
small times. The invariant measure is d = e;U dx (where dx is Lebesgue
measure on Rn) and ;(f f ) = jrf j2. The prime example is U (x) = jxj2
which is associated to the Ornstein-Uhlenbeck operator. This operator is
however only hypercontractive and not ultracontractive (i.e. sup pt (x y ) is
unbounded). We will actually be concerned with functions U that growth
faster than jxj2 at innity. Although we work for simplicity with the Laplace
operator on Rn, the results would hold similarly for an arbitrary generator
satisfying a logarithmic Sobolev inequality (1.5) in the abstract setting (and
with jrU j2 replaced by ;(U )).
The next proposition transfers the optimal (LS) inequality for dx into a
logarithmic Sobolev inequality for . It is the key argument in the applications.
R
Proposition 3.1. For every smooth f on Rn with f 2 d = 1 and every
v > 0,
Z
where
Z
2
n
2
f log f d 2 log v ; 1 + nev jrf j d + c(v )
2
2
c(v ) = sup 41ev 2U (x) ; jrU j2(x) + U (x) :
x2R
n
Proof. The proof is easy. We start with inequality (1.2)
Z
2 Z
n
2
g log g dx 2 log ne jrg j dx
2
2
R
where g 2dx = 1. By concavity of the log function, for every v > 0,
Z
Z
n
2
2
2
2
g log g dx 2 log v ; 1 + n ev jrg j dx :
R
Set now g 2 = f 2 e;U . Then f 2 d = 1 and
Z
Z
Z
Z
jrgj2dx = jrf j2d ; f rf rUd + 14 f 2 jrU j2d:
It follows that
Z
Z
n
2
2
2
2
f log f d 2 log v ; 1 + nev jrf j d
Z + f 2 41ev 2U ; jrU j2 + U d:
Proposition 3.1 is established.
tu
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
406
D. BAKRY, D. CONCORDET, M. LEDOUX
Assume now we can nd a concave increasing function : R+ ! R+ such
that for every u 0 there exists v > 0 with
n log v ; 1 + 2u + c(v ) (u):
(3:1)
2
nev
Then, by Proposition 3.1,
Z
Z
f log f d 2
2
jrf j d
2
R
for every smooth f on Rn with f 2 d = 1. We may then make use of the
general theory recalled in Section 1 to obtain bounds on the heat kernel of
L. Rather than to state a general result, let us inspect a number of examples
drawn from Kavian et al. (1993)] (see also Rosen (1976)]).
Take, to start with, U (x) = jxj, > 2. Then, we may take
c(v ) Cv =(;2) v 1
where C > 0 only depends on n and and may vary from line to line below.
The function
;
(u) = C 1 + u=(2;2) u 0
will then satisfy (3.1). As a consequence, the bound (1.6) yields, for every
t > 0,
C
kPtk11 exp =(;2) :
t
Similarly, if U (x) = jxj2 (log(1 + jxj2 )), > 1, we obtain
kPtk11 exp Ct=(;1) exp 1=(C;1) :
t
An interesting limiting case is obtained when = 1 in the last example.
It is known from Kavian et al. (1993)] that the associated semigroup is
not ultracontractive, but what is called immediately hypercontractive (i.e.
kPtk11 = 1 but kPtkpq < 1 for every t > 0, 1 < p < q < 1) . This is
described by the bound
r
2
1
Ct
C
log
r
kPtk2q exp 1 ; r log2 r exp t
where q > 2, r = 4q 2 =(q ; 1), which is obtained by the same method.
Acknowledgement
We thank L. Salo-Coste for helpful comments leading to Corollary 1.6.
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407
OPTIMAL HEAT KERNEL BOUNDS
407
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D. Bakry, M. Ledoux: Departement de Mathematiques, Laboratoire de Statistique et Probabilites associe au C.N.R.S., Universite Paul Sabatier, 31062
Toulouse, France. Email: [email protected], [email protected]
cole Veterinaire de Toulouse, 31067
D. Concordet: Unite de Biometrie, E
Toulouse, France et Departement de Mathematiques, Laboratoire de Statistique et Probabilites associe au C.N.R.S., Universite Paul Sabatier, 31062
Toulouse, France. Email: [email protected]
ESAIM: P&S, December 1997, Vol. 1, pp. 391{407