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ANNALES DE
L’INSTITUT FOURIER
Université Grenoble Alpes
Les Annales de l’institut Fourier sont membres du
Centre Mersenne pour l’édition scientique ouverte
www.centre-mersenne.org
François Ledrappier & Lin Shu
Corrigendum to “Dierentiating the stochastic entropy in
negatively curved spaces under conformal changes”
Tome 71, no 1 (2021), p. 89-96.
<http://aif.centre-mersenne.org/item/AIF_2021__71_1_89_0>
© Association des Annales de l’institut Fourier, 2021,
Certains droits réservés.
Cet article est mis à disposition selon les termes de la licence
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Ann. Inst. Fourier, Grenoble
71, 1 (2021) 89-96
CORRIGENDUM TO “DIFFERENTIATING THE
STOCHASTIC ENTROPY IN NEGATIVELY CURVED
SPACES UNDER CONFORMAL CHANGES”
Annales de l’institut Fourier, vol. 67 (2017), n°3, 1115–1183
by François LEDRAPPIER & Lin SHU
Abstract. — We correct an error in [5, Proposition 2.16], and explain the sub-
sequent changes in the proof of the main results.
Résumé. — Nous corrigeons une erreur dans [5, Proposition 2.16] et nous dé-
taillons les modifications à apporter aux démonstrations des principaux résultats.
There is an error in our paper [5, Proposition 2.16], in the expression of
the entropy of a foliated diffusion due to our dismissing the non-symmetry
of the diffusions. The main results on the regularities of the entropy (Propo-
sition 3.2 and Theorem 5.1(ii)) and the structure of their proofs are not
affected. But in several places, the Martin kernels we used should be re-
placed by the Martin kernels of the formal adjoint operator and the details
of the first step of the proof of Theorem 5.1(ii) should be corrected.
1. Setting
In this section, we briefly recall the settings and the notations from [5].
We locate the error and introduce the new notations we use.
Let (M, g) be an m-dimensional closed Riemannian manifold with neg-
ative curvature, where g is C3, (M̃, g̃) the universal cover and G the fun-
damental group of M that acts freely on M̃ in such a way that M can be
identified with a space of orbits of this action. We choose once for all a
connected fundamental domain M0 for the action of G on M̃.
Keywords: formal adjoint, Martin kernel, stochastic entropy.
2020 Mathematics Subject Classification: 37D40, 58J65.
90 François LEDRAPPIER & Lin SHU
The unit tangent bundles of M and M̃ , denoted SM and SM̃ , can be
identified with M0 × ∂M̃ and M̃ × ∂M̃ , respectively, where ∂M̃ is the
geometric boundary of M̃ . For each v = (x, ξ) ∈ SM̃ , its stable manifold
with respect to the geodesic flow on SM̃ , denoted W s(v), is the collection
of initial vectors w of geodesics asymptotic to ξ and can be identified with
M̃×{ξ}. The collection ofW s(v) defines the stable foliation (lamination)W
on SM , whose leaves are discrete quotients of M̃ and are naturally endowed
with the Riemannian metric induced from g̃. A differential operator L on
(the smooth functions on) SM with continuous coefficients is said to be
subordinate to the stable foliation W, if for every smooth function F on
SM the value of L(F ) at v ∈ SM only depends on the restriction of F to
the leaf of W that contains v. Given such an operator L, a Borel measure
m on SM is called L-harmonic if it satisfies, for every smooth function f
on SM , ∫
L(f) dm = 0.
In this note, we are interested in the case L = ∆ + Y , where Y is a
section of the tangent bundle of W over SM of class Ck,αs for some k > 1
and α ∈ [0, 1) such that Y ∗, the dual of Y in the cotangent bundle to
W over SM satisfies dY ∗ = 0 leafwisely and such that the Pressure of
the function −〈X,Y 〉 is positive (the sign convention for the argument of
the Pressure function in [2] is opposite to the usual one), where X is the
geodesic spray and can be considered as a section of the tangent bundle
of W over SM . Such a L has a unique L-harmonic probability measure m
([2, Theorem A]).
Extend L to a G-equivariant operator on SM̃ = M̃ ×∂M̃ which we shall
denote with the same symbol. The operator L defines a Markovian family
of probabilities on Ω̃+, the space of continuous paths of ω̃ : [0,+∞) →
SM̃ , equipped with the smallest σ-algebra A for which the projections
Rt : ω̃ 7→ ω̃(t) are measurable. Indeed, for v = (x, ξ) ∈ SM̃ , let Lv denote
the laminated operator of L onW s(v). It can be regarded as an operator on
M̃ with corresponding heat kernel functions pv(t, y, z), t ∈ R+, y, z ∈ M̃ .
Define
p(t, (x, ξ), d(y, η)) = pv(t, x, y)dVol(y)δξ(η),
where δξ( · ) is the Dirac function at ξ. Then the diffusion process on
W s(v) with infinitesimal operator Lv is given by a Markovian family
{Pw}w∈M̃×{ξ}, where for every t > 0 and every Borel set A ⊂ M̃ × ∂M̃ we
have
Pw ({ω̃ : ω̃(t) ∈ A}) =
∫
A
p(t,w, d(y, η)).
ANNALES DE L’INSTITUT FOURIER
CORRIGENDUM 91
The group G acts naturally and discretely on the space Ω̃+ with quotient
the space Ω+ of continuous paths in SM , and this action commutes with
the shift. Therefore, the measure P̃ =
∫
Pv dm̃(v) on Ω̃+ is the extension
of a shift invariant ergodic probability measure P on Ω+. Elements in Ω+
will be denoted by ω, which can be identified with its lift in Ω̃+ starting
from M0. By [1], for each v ∈ SM̃, there exists the corresponding Green
function Gv( · , · ) on M̃ × M̃ . We define the Green function G( · , · ) on
SM̃ × SM̃ as being
G ((y, η), (z, ζ)) := G(y,η)(y, z)δη(ζ), for (y, η), (z, ζ) ∈ SM̃,
where δη is the Dirac function at η. Moreover, for x, y ∈ M̃ and η ∈ ∂M̃,
the following limit exists and defines the Martin kernel kv(x, y, η):
kv(x, y, η) = lim
z→η
G(x,ξ)(x, z)
G(x,ξ)(y, z)
.
By [2], Corollary B7, (x, ξ) 7→ ∇y log k(x,ξ)(x, y, ξ)|y=x is a Hölder continu-
ous function.
Denote by m̃ the G-invariant measure which extends m on M̃ × ∂M̃ .
The (stochastic) entropy hL of L (introduced by Kaimanovich ([3, 4])) is
given by
hL := lim
t→+∞
−1
t
∫
M0×∂M̃
(log p(t, (x, ξ), (y, η))) p(t, (x, ξ), d(y, η)) dm̃(x, ξ).
We have the following limit expressions for hL:
Proposition 1.1 ([5, Proposition 2.4]). — For P-a.e. paths ω ∈ Ω+,
hL = lim
t→+∞
−1
t
log p(t, ω(0), ω(t))
= lim
t→+∞
−1
t
log G(ω(0), ω(t)).(1.1)
Our attempt in [5, Proposition 2.16], is to replace G in (1.1) by the
Martin kernel and use it to find an integral expression. In its computation,
we replaced at some point Gv(y,z)Gv(y,x) , for y far away in M̃ , by kv(x, z, ξ)
whereas that ratio is close to the Martin kernel k∗v(x, z, ξ) of the formal
adjoint L∗v, which we define as follows. For v ∈ SM̃, let L∗v be the formal
adjoint of Lv, i.e., the operator such that for all ϕ,ψ ∈ C∞c (M̃),∫
M̃
ϕL∗vψ dVolg̃ =
∫
M̃
(Lvϕ)ψ dVolg̃.
The operator L∗ subordinate to W given by L∗ψ = ∆ψ−Div(ψY ) admits
L∗v as its laminated operator on W s(v). Observe that the operator L∗
TOME 71 (2021), FASCICULE 1
92 François LEDRAPPIER & Lin SHU
is also weakly coercive and that its Green function G∗(x, y) is given by
G∗v(x, y) = Gv(y, x). Again, for v = (x, ξ) ∈ SM̃ , y ∈ M̃ and η ∈ ∂M̃, the
following limit exists and defines the Martin kernel k∗v(x, y, η):
(1.2) k∗v(x, y, η) = lim
z→η
G∗(x,ξ)(x, z)
G∗(x,ξ)(y, z)
= lim
z→η
G(x,ξ)(z, x)
G(x,ξ)(z, y)
.
By [2], Corollary B7, the function (x, ξ) 7→ ∇y log k∗(x,ξ)(x, y, ξ)|y=x is
Hölder continuous. Moreover, if we write Y = ∇ log fv for some C2 func-
tion fv, we have L∗vψ = fvLv
(
(fv)−1ψ
)
. Thus, ψ 7→
(
(fv)−1ψ
)
is a one-
to-one correspondence between L∗v-harmonic functions and Lv-harmonic
functions, so that, for all v = (x, ξ) ∈ SM̃ , y ∈ M̃ , η ∈ ∂M̃ ,
(1.3) k∗v(x, y, η) =
fv(y)
fv(x)
kv(x, y, η).
In [2, Appendix B], Hamenstädt studied the operator L∗v and its Martin
kernel k∗v(x, z, ξ) and she showed that they enjoy many properties of the
operator L and its Martin kernel kv(x, z, ξ). In particular, our argument
for Proposition 3.2 still holds after correcting kv by k∗v. Finally, to get
Theorem 5.1(ii), we apply the above to operators which come from for-
mally self-adjoint operators, it is not surprising that the final formula is
the same as the one obtained when overlooking the lack of symmetry of
the intermediate operator.
2. Stochastic entropy for laminated diffusions
With the above notations, the following replaces Proposition 2.16 in [5]:
Proposition 2.1. — Let L = ∆ + Y be such that Y ∗, the dual of Y
in the cotangent bundle to the stable foliation over SM , satisfies dY ∗ = 0
leafwisely and pr(−〈X,Y 〉) > 0. Then,
hL = −
∫
M0×∂M̃
(∆ + Y )y
(
log k∗(x,ξ)(y, ξ)
) ∣∣∣
y=x
dm̃(x, ξ)(2.1)
=
∫
M0×∂M̃
(
‖∇y log k(x,ξ)(y, ξ)|y=x‖2 −Div Y − ‖Y ‖2
)
dm̃(x, ξ).(2.2)
Proof. — The proof of (2.1) copies the proof of Proposition 2.16 in [5],
up to the point where one uses (1.2). For P-a.e. path ω ∈ Ω+, we still
denote ω its projection to M̃ and write v := ω(0). When t goes to infinity,
we see that
lim sup
t→+∞
|logGv(x, ω(t))− log k∗v(ω(t), ξ)| < +∞.
ANNALES DE L’INSTITUT FOURIER
CORRIGENDUM 93
Indeed, let zt be the point on the geodesic ray γω(t),ξ closest to x. Then, as
t→ +∞,
Gv(x, ω(t))  Gv(zt, ω(t)) 
Gv(y, ω(t))
Gv(y, zt)
for all y on the geodesic going from ω(t) to ξ with d(y, ω(t)) > d(y, zt) + 1,
where  means up to some multiplicative constant independent of t. The
first  comes from Harnack inequality using the fact that supt d(x, zt) is
finite P-almost everywhere. (For P-a.e. ω ∈ Ω+, η = limt→+∞ ω(t) differs
from ξ and d(x, zt), as t → +∞, tends to the distance between x and the
geodesic asymptotic to ξ and η in opposite directions.) The second  comes
from Ancona’s inequality ([1]). Replace Gv(y, ω(t))/Gv(y, zt) by its limit
as y → ξ, which is k∗(zt,ξ)(ω(t), ξ) by (1.2), which is itself  k
∗
(x,ξ)(ω(t), ξ)
by Harnack inequality again.
Since the L-diffusion has leafwise infinitesimal generator ∆ + Y and is
ergodic, we obtain
hL = lim
t→+∞
−1
t
∫ t
0
∂
∂s
(log k∗(x,ξ)(ω(s), ξ)) ds
= − lim
t→+∞
1
t
∫ t
0
(∆ + Y )y
(
log k∗(x,ξ)(y, ξ)
) ∣∣∣
y=ω(s)
ds
= −
∫
M0×∂M̃
(∆ + Y )y
(
log k∗(x,ξ)(y, ξ)
) ∣∣∣
y=x
dm̃(x, ξ).
For proving (2.2), one starts by using (1.3) to write
hL = −
∫
M0×∂M̃
(∆ + Y )y
(
log k(x,ξ)(y, ξ) + log f(x,ξ)(y)
) ∣∣
y=x dm̃(x, ξ).
Since the Martin kernel k(x,ξ)( · , ξ) satisfies (∆ + Y )y(k(x,ξ)(y, ξ))|y=x = 0,
(∆ + Y )y
(
log k(x,ξ)(y, ξ)
)
|y=x = −‖∇y
(
log k(x,ξ)(y, ξ)
)
|y=x‖2.
On the other hand, since Y = ∇y log f(x,ξ)|y=x, we have
(∆ + Y )y
(
log f(x,ξ)(y)
)
|y=x = Div Y + ‖Y ‖2.
Relation (2.2) follows. 
For the proof of Proposition 3.2 in [5], the places that are affected by
Proposition 2.16 and should be corrected are as follows: the u1,Z1t and σ1
of page 1141 have to be replaced by u∗1,Z
1,∗
t and σ∗1 for k∗. For Lemma 3.11,
the changes are only formal, replacing k0v(ω(s), ξ) by k
0,∗
v (ω(s), ξ) on
page 1160 and in the estimate of I3.
TOME 71 (2021), FASCICULE 1
94 François LEDRAPPIER & Lin SHU
3. The entropy part regularity
Let λ ∈ (−1, 1) 7→ gλ = e2ϕλg be a C3 curve in the space of C3 negatively
curved metrics with g0 = g and constant volume. Denote g̃λ the G-invariant
extension of gλ to M̃ . Its geometric boundary for M̃ can be identified with
∂M̃ of g̃. For each gλ, let ∆λ denote the laminated Laplacian for its stable
foliation (and also the lift to {M̃×{ξ}}ξ∈∂M̃ ), mλ the unique ∆λ-harmonic
probability measure with G-invariant lift m̃λ to M̃ × ∂M̃ , hλ the entropy
of ∆λ and kλv the leafwise Martin kernels for ∆λ. Note that the formal
adjoint of ∆λ is itself and hence has the same Martin kernel functions as
∆λ. By [3] (see also (2.1)),
hλ = −
∫
M0×∂M̃
∆λy
(
log kλ(x,ξ)(y, ξ)
) ∣∣∣
y=x
dm̃λ(x, ξ).
Let L̂λ := ∆ + Zλ with Zλ = (m− 2)∇ϕλ ◦ φ, where φ : SM 7→ M is the
projection. Then
(3.1) ∆λ = e−2ϕ
λ◦φL̂λ.
Leafwisely, Zλ is the dual of the closed form (m− 2)dϕλ ◦φ. Moreover, the
pressure of the function −〈X,Z0〉 = 0 is positive. Therefore, there exists
δ > 0 such that for |λ| < δ, the pressure of the function −〈X,Zλ〉 is still
positive. For such λ, we denote m̂λ for the G-invariant extension to SM̃
of the harmonic probability measure corresponding to L̂λ with respect to
metric g. We denote k̂λv the Martin kernels for L̂λ. Let L̂λ,∗ be the formal
adjoint of L̂λ and its Martin kernel functions are denoted by k̂λ,∗v . Let ĥλ
be the entropy for L̂λ. We have by Proposition 2.1 that
ĥλ = −
∫
M0×∂M̃
L̂λy
(
log k̂λ,∗(x,ξ)(y, ξ)
) ∣∣∣
y=x
dm̂λ(x, ξ).
The proof of Theorem 5.1(ii) in [5] concerning the differential formula of
hλ in λ was divided into two parts, where the first part is to show
(I)h := lim
λ→0
1
λ
(hλ − ĥλ) = 0
and the other is to analyze the differential of ĥλ in λ. The proof of (I)h = 0
is affected by the use of the wrong formula of [5, Proposition 2.16]. We
correct it as follows.
Claim 3.1. — We have (I)h = 0.
ANNALES DE L’INSTITUT FOURIER
CORRIGENDUM 95
Proof. — For λ with |λ| < δ, set Vλ :=
∫
M0×∂M̃ e
−2ϕλ◦φ dm̃λ. We show
ĥλ = V −1λ hλ. Since the volume of gλ is constant, limλ→0
1
λ (Vλ − 1) = 0,
and the claim follows.
Using (3.1), we obtain that L̂λ,∗α = 0 iff ∆λ
(
e−(m−2)ϕ
λ◦φα
)
= 0. Con-
sequently, we have
k̂λ,∗v = e(m−2)ϕ
λ◦φkλv.
Using dm̂λ = V −1λ e−2ϕ
λ◦φ dm̃λ and (3.1), we compute that
Vλĥλ = −
∫
M0×∂M̃
∆λy
(
log k̂λ,∗(x,ξ)(y, ξ)
) ∣∣∣
y=x
dm̃λ(x, ξ)
= −
∫
M0×∂M̃
∆λ
(
log(kλ(x,ξ)(y, ξ)
) ∣∣∣
y=x
dm̃λ(x, ξ)
− (m− 2)
∫
M0×∂M̃
∆λ(ϕλ ◦ φ) dm̃λ
= −
∫
M0×∂M̃
∆λ
(
log(kλ(x,ξ)(y, ξ)
) ∣∣∣
y=x
dm̃λ(x, ξ) = hλ,
where the third equality holds since mλ is ∆λ harmonic. 
For the computation of the derivative, we have to replace u1 by u∗1. Since
in our case, L̂0 = ∆ is formally self-adjoint, u∗1 = u1 and our previous
computation is unchanged.
BIBLIOGRAPHY
[1] A. Ancona, “Negatively curved manifolds, elliptic operators, and the Martin bound-
ary”, Ann. Math. 125 (1987), no. 3, p. 495-536.
[2] U. Hamenstädt, “Harmonic measures for compact negatively curved manifolds”,
Acta Math. 178 (1997), no. 1, p. 39-107.
[3] V. A. Kăimanovich, “Brownian motion and harmonic functions on covering mani-
folds. An entropic approach”, Dokl. Akad. Nauk SSSR 288 (1986), no. 5, p. 1045-
1049.
[4] ———, “Brownian motion on foliations: entropy, invariant measures, mixing”,
Funkts. Anal. Prilozh. 22 (1988), no. 4, p. 82-83, Russian, translation in Funct.
Anal. Appl. 22 (1988), no. 4, p. 326-328.
[5] F. Ledrappier & L. Shu, “Differentiating the stochastic entropy for compact nega-
tively curved spaces under conformal changes”, Ann. Inst. Fourier 67 (2017), no. 3,
p. 1115-1183.
Manuscrit reçu le 31 janvier 2019,
accepté le 3 août 2020.
François LEDRAPPIER
Sorbonne Université, UMR 8001, LPSM
Boîte Courrier 158
4, Place Jussieu
TOME 71 (2021), FASCICULE 1
96 François LEDRAPPIER & Lin SHU
75252 PARIS cedex 05 (France)
fledrapp@nd.edu
Lin SHU
LMAM, School of Mathematical Sciences
Peking University
Beijing 100871 (China)
lshu@math.pku.edu.cn
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