Eulerian calculus for the displacement convexity in the Wasserstein distance

0.48 MB·English

by  Sara Daneri

Eulerian calculus for the displacement convexity in the
Wasserstein distance
Sara Daneri Giuseppe SavarØ
8
0 S.I.S.S.A., Trieste ∗ Universit(cid:224) di Pavia †
0
2 January 15, 2008

n
a
J
Abstract
6
1 Inthispaperwegiveanewproofofthe(strong)displacementconvexityofaclassofinte-
gralfunctionalsde(cid:28)nedonacompactRiemannianmanifoldsatisfyingalowerRiccicurvature

] bound. Our approach does not rely on existence and regularity results for optimal transport
P
maps on Riemannian manifolds, but it is based on the Eulerian point of view recently intro-
A ducedbyOtto-Westdickenbergin[19]andonthemetriccharacterizationofthegradient
. (cid:29)ows generated by the functionals in the Wasserstein space.
h
t Keywords: Gradient(cid:29)ows,displacementconvexity,heatandporousmediumequation,nonlinear
a
m di(cid:27)usion,optimaltransport,Kantorovich-Rubinstein-Wassersteindistance,Riemannianmanifolds
with a lower Ricci curvature bound.
[

1
1 Introduction
v
5
5 In this paper we give a new proof, based on a gradient (cid:29)ow approach and on the Eulerian point
4
of view introduced by [19], of the so called (cid:16)displacement convexity(cid:17) for integral functionals as
2
(cid:90)
. dµ
1 E(µ):= e(ρ)dV+e(cid:48)(∞)µ⊥(M), ρ= , (1.1)
0 M dV
8
where µ is a Borel probability measure on a compact, connected Riemannian manifold without
0
: boundary(M,g),VisthevolumemeasureonMinducedbythemetrictensorg,µ⊥ isthesingular
v part of µ with respect to V, e:[0,+∞)→R is a smooth convex function satisfying the so called
i
X McCann conditions (see (1.7) below), and e(cid:48)(∞)= lim e(r). When e has a superlinear growth,
r→+∞ r
r e(cid:48)(∞)=+∞ so that µ should be absolutely continuous with respect to V when E(µ) is (cid:28)nite.
a
Displacement convexity for integral functionals. Thenotionofdisplacementconvexity has
been introduced by McCann [15] to study the behavior of integral functionals like (1.1) along
optimal transportation paths, i.e. geodesics in the space of Borel probability measures P(M)
endowed with the L2-Kantorovich-Rubinstein-Wasserstein distance.
Recall that (the square of) this distance can be de(cid:28)ned by the following optimal transport
problem
(cid:110)(cid:90)
W2(µ0,µ1):=min d2(x,y)dσ(x,y):σ ∈P(M×M),
2
M×M (1.2)
(cid:111)
σ(M×B)=µ0(B), σ(B×M)=µ1(B) ∀B Borel set in M ,
∗S.I.S.S.A.,ViaBeirut2-4,34014,Trieste,Italy.
e-mail: [email protected].
†DepartmentofMathematics,ViaFerrata1,27100,Pavia,Italy.
e-mail: [email protected],
web: http://www.imati.cnr.it/∼savare/
1
forthecostfunctioninducedbytheRiemanniandistancedonthemanifoldM. Wekeeptheusual
notation to denote by P (M) the metric space (P(M),W ), that is called Wasserstein space;
2 2
being M compact, W induces the topology of the weak convergence of probability measures (i.e.,
2
the weak∗ topology associated to the duality of P(M) with C0(M)).
As in any metric space, (minimal, constant speed) geodesics can be de(cid:28)ned as curves µ : s ∈
[0,1](cid:55)→µs ∈P (M) between µ0 and µ1 satisfying
2
W (µr,µs)=|s−r|W (µ0,µ1) ∀ 0≤r ≤s≤1. (1.3)
2 2
A functional E :P(M)→(−∞,+∞] is then (strongly) displacement convex (or, more generally,
displacement λ-convex for some λ∈R) if, for all Wasserstein geodesics {µs} ⊂P (M), we
0≤s≤1 2
have
λ
E(µs)≤(1−s)E(µ0)+sE(µ1)− s(1−s)W2(µ0,µ1), ∀s∈[0,1]. (1.4)
2 2
A weaker notion is also often considered: one can ask that there exists at least one geodesic
connecting µ0 to µ1 along which (1.4) holds.
The term (cid:16)displacement convexity(cid:17) arises from the strictly related concept of (cid:16)displacement
interpolation(cid:17) introduced by [15] in the Euclidean case M = Rd; in a general metric setting,
property (1.4) is simply called, as in the Riemannian case, (cid:16)λ−geodesic convexity(cid:17) (or (cid:16)geodesic
convexity(cid:17) if λ=0).
It is possible to show [4] that the measures µs can also be de(cid:28)ned through the formula
µs(B):=σ(cid:0){(x,y)∈Rd×Rd :(1−s)x+sy ∈B}(cid:1), where σ is a minimizer of (1.2). (1.5)
AsimilarconstructioncanalsobeperformedinaRiemannianmanifold[14,20,13]: thesegments
s (cid:55)→ (1−s)x+sy should be substituted by a Borel map γ : M×M → C0([0,1];M) that at
each couple (x,y) ∈ M×M associate a (minimal, constant speed) geodesic s (cid:55)→ γs(x,y) in M
connecting x to y. We have the representation formula
µs(B):=σ(cid:0){(x,y)∈M×M:γs(x,y)∈B}(cid:1), where σ is a minimizer of (1.2). (1.6)
Afterthepioneeringpaper[15],thenotionofdisplacementconvexityforintegralfunctionalsfound
applications in many di(cid:27)erent (cid:28)elds, as Functional inequalities [18, 2, 9], generation, contraction,
andasymptoticpropertiesofdi(cid:27)usionequationsandGradient(cid:29)ows[17,1,19,4,8,5],Riemannian
Geometry and synthetic study of Metric-Measure spaces [20, 14].
In the context of Riemannian manifolds it turns out that displacement λ-convexity of certain
classesofentropyfunctionalsisequivalenttoalowerboundfortheRiccicurvatureofthemanifold.
TheconnectionbetweendisplacementconvexityandRiccicurvature,introducedby[18],wasthen
further deeply studied by [18, 9, 10, 20]; the equivalence has been proved by Sturm and Von
Renesse in [23], who considered the case in which the domain of the functional consists only of
measures that are absolutely continuous with respect to the volume measure, and then completed
by Lott and Villani [14] (with the remarks made in [12], where convexity in the strong form has
been proved), who extended the previous results to the functionals de(cid:28)ned by (1.1) on all P(M).
We refer to the forthcoming monograph [22] for further references, details, and discussions.
The strategy followed by the authors of [9] (and by all the following contributions) in order
to characterize the displacement convexity of entropy functionals relies on a characterization of
optimaltransportationandWassersteingeodesics[16]andonacarefulstudyoftheJacobianpro-
pertiesoftheexponentialfunctionwhicharecrucialtoestimatetheintegralfunctionalsalongthis
class of curves. The lack of regularity of Wasserstein geodesics and the lack of global smoothness
of the squared distance function d2 on the manifold M (due to the existence of the cut-locus)
require a careful use of non-smooth analysis arguments and non trivial approximation processes
to extend the results to geodesics between arbitrary measures (see [14, 12]).
The main result is the following
2
Theorem 1.1 (I) If e∈C∞(0,+∞) satis(cid:28)es the McCann conditions:
(cid:18) (cid:19)
U(ρ):=ρe(cid:48)(ρ)−(cid:0)e(ρ)−e(0 )(cid:1)≥0, ρU(cid:48)(ρ)− 1− 1 U(ρ)≥0, n:=dim(M)>1 (1.7)
+
n
and M has nonnegative Ricci curvature, then the functional E de(cid:28)ned by (1.1) is (strongly)
displacement convex.
(II) If E is the relative entropy functional, corresponding to e(ρ)=ρlogρ (which satis(cid:28)es (1.7)
in any dimension) in (1.1), and there exists λ∈R such that
Ric (ξ,ξ)≥λ(cid:104)ξ,ξ(cid:105) ∀x∈M, ∀ξ ∈T M, (1.8)
x gx x
then the functional E de(cid:28)ned by (1.1) is (strongly) displacement λ-convex.
Remark 1.2 Besides the logarithmic entropy corresponding to e(ρ) = ρlogρ (and U(ρ) = ρ),
typical examples of functionals that satisfy properties (1.7) are
e(ρ)= 1 ρm, U(ρ)=ρm, m≥1− 1. (1.9)
m−1 n
Werecallthatassumptions(1.7)implytheconvexityofthefunctionρ(cid:55)→e(ρ)(sincethedimension
n is greater than 1, they are in fact more restrictive).
Aim of the paper: an Eulerian approach to displacement convexity. In this paper we
present an alternative proof of Theorem 1.1, which does not rely on the existence and smoothness
of optimal transport maps and geodesics for the Wasserstein distance.
Our strategy can be described in three steps:
1. Following the approach suggested by Otto-Westdickenberg in [19], we work in the sub-
spacePar(M)ofmeasureswithsmoothandpositivedensitiesandweusethe(cid:16)Riemannian(cid:17)
2
formula for the Wasserstein distance, originally introduced in the Euclidean framework by
Benamou-Brenier [6]: if µi =ρiV∈Par(M), i=0,1, then [19, Prop. 4.3]
2
(cid:110)(cid:90) 1(cid:90) (cid:111)
W2(µ0,µ1)= inf |∇φs|2ρsdVds ∀µ0,µ1 ∈Par(M) (1.10)
2 C(µ0,µ1) 0 M 2
where
(cid:110)
C(µ0,µ1)= (ρ,φ):ρ∈C∞([0,1]×M;R ), φ∈C∞([0,1]×M)
+
(1.11)
(cid:111)
∂ ρs+∇·(ρs∇φs)=0 in (0,1)×M, µi =ρiV .
s
Even though the Wasserstein space can’t be endowed with a smooth Riemannian structure,
(1.11) still shows a (cid:16)Riemannian(cid:17) characterization of the Wasserstein distance on Par(M).
2
2. The second important fact, originally showed by the so-called (cid:17)Otto calculus(cid:17) in [17], is that
the nonlinear di(cid:27)usion equation
∂tρt−∆gU(ρt)=0 in [0,+∞)×M, ρ|t=0 =ρ0, (1.12)
where U :R+ →R is the function de(cid:28)ned in (1.7) and ∆ is the Laplace-Beltrami operator
g
on M, is the gradient (cid:29)ow of the functional (1.1) in P (M). Indeed, (1.12) corresponds to
2
the heat equation if U is the logarithmic entropy and to the porous medium equation if U
is de(cid:28)ned by (1.9).
Starting directly from (1.10) and owing to the fact that the (cid:29)ow generated by (1.12) pre-
serves smooth and positive densities, when Ric(M) ≥ 0 we shall show that the measures
µ = ρ V ∈ Par(M) associated to the solutions of (1.12) also solve the Evolution Varia-
t t 2
tional Inequality (E.V.I.)
1d+
W2(ν,µ )≤E(ν)−E(µ ) ∀t≥0, ν ∈Par(M), (1.13)
2 dt 2 t t 2
3
which has been introduced in [4] as a purely metric characterization of the gradient (cid:29)ows of
geodesically convex functionals in metric spaces (and in particular in P (Rd)); here
2
d+ ζ(t+h)−ζ(t)
ζ(t)=limsup (1.14)
dt h
h↓0
for every real function ζ :[0,+∞)→R.
When Ric(M) ≥ λ (a shorthand for (1.8)), we also show that the solutions of the heat
equation satisfy the modi(cid:28)ed inequality
1d+ λ
W2(ν,µ )+ W2(ν,µ )≤E(ν)−E(µ ) ∀t≥0, ν ∈Par(M), (1.15)
2 dt 2 t 2 2 t t 2
where E is the relative entropy functional whose integrand function is e(ρ) = ρlogρ. Note
that (1.15) reduces to (1.13) when λ = 0. In order to prove (1.13) and (1.15), we propose
an (cid:16)Eulerian(cid:17) strategy which could be adapted to more general situations.
3. The third crucial fact is the following: whenever a functional E satis(cid:28)es (1.13) (or, more
generally, (1.15)) for a given semigroup : µ = ρ V (cid:55)→ µ = ρ V in Par(M), E is dis-
t 0 0 t t 2
placement convex (resp. displacement λ-convex). Thus the question of the behavior of E
along geodesics can be reduced to a di(cid:27)erential estimate of E along the smooth and positive
solutions of its gradient (cid:29)ow.
Plan of the paper. In Section 2 we present the main ideas of our approach in the simpli(cid:28)ed
((cid:28)nite-dimensionalandsmooth)settingofgeodesicallyconvexfunctionsonRiemannianmanifolds.
We think that these ideas are su(cid:30)ciently general to be useful in other circumstances, at least for
distances which admits a Riemannian characterization as (1.10), see e.g. [11, 7]
After a brief review of the de(cid:28)nition of (gradient) λ-(cid:29)ows in arbitrary metric spaces (basically
following the ideas of [4]), we present in Section 3 our (cid:28)rst result, showing that the existence of a
(cid:29)ow satisfying the E.V.I. (1.15) (even on a dense subset of initial data, such as Par(M)) entails
2
the (strong) displacement λ-convexity of the functional E.
Following the strategy explained in the second section, in the last two sections we prove the
di(cid:27)erential estimates showing that (1.12) satis(cid:28)es (1.13) (in Section 4) or, in the case of the Heat
equation, (1.15) (in Section 5).
2 Gradient (cid:29)ows and geodesic convexity in a smooth setting
Contraction semigroups and action integrals. In order to explain the main point of our
strategy, let us (cid:28)rst consider the simple setting of a smooth function F : X → R on a com-
plete Riemannian manifold X with metric (cid:104)·,·(cid:105) , (squared) norm |ξ|2 = (cid:104)ξ,ξ(cid:105) , and the endowed
g g g
Riemannian distance
(cid:110)(cid:90) 1(cid:12) (cid:111)
d2(u,v):=min (cid:12)γ˙s|2ds, γ :[0,1]→X, γ0 =v, γ1 =u . (2.1)
g
0
In a smooth setting, the geodesic λ-convexity of F can be expressed through the di(cid:27)erential
condition
d2
F(γs)≥λ|γ˙s|2 (2.2)
ds2 g
along any geodesic curve γ minimizing (2.1). As we discussed in the introduction, the direct
computationof (2.2)couldbedi(cid:30)cultinanon-smooth,in(cid:28)nitedimensionalsetting;itistherefore
important to (cid:28)nd equivalent conditions which avoid twofold di(cid:27)erentiation along geodesics. One
possibility, suggested in [19], is to (cid:28)nd equivalent conditions to geodesic λ-convexity in terms of
the gradient (cid:29)ow generated by F.
4
Let us recall that the gradient (cid:29)ow of F is a continuous semigroup of (time-dependent) maps
S : X → X, t ∈ [0,+∞), which at every initial datum u associate the curve u := S (u) solution
t t t
of the di(cid:27)erential equation
u˙ =−∇F(u ) ∀t≥0, u =u. (2.3)
t t 0
It is well known that, when F is geodesically λ-convex, S is λ-contracting, i.e.
t
d2(S (u),S (v))≤e−2λtd2(u,v) ∀u,v ∈X. (2.4)
t t
By the semigroup property, (2.4) is also equivalent to the di(cid:27)erential inequality (see (1.14))
d+ (cid:12)
d2(S (u),S (v))(cid:12) ≤−2λd2(u,v) ∀u,v ∈X. (2.5)
dt t t (cid:12)t=0
[19]revertsthisargumentandobservesthatitcouldbeeasiertodirectlyprove(2.5)byadi(cid:27)erential
estimate involving only the action of the semigroup along smooth curves; as a byproduct, one
should obtain the convexity of F. To this aim, they consider a smooth curve γs, s ∈ [0,1],
connecting v to u, and the action integral A associated to its smooth perturbation
t
(cid:90) 1
γts :=St(γs), Ast :=(cid:12)(cid:12)∂sγts(cid:12)(cid:12)2g, At := Astds, (2.6)
0
where ∂ γ,∂ γ denotes the tangent vectors in T X obtained by di(cid:27)erentiating w.r.t. s and t re-
s t γ
spectively. Since, by the very de(cid:28)nition of d,
d2(S (v),S (u))≤A (2.7)
t t t
and for every ε>0 one can always (cid:28)nd a curve γs so that A ≤d2(u,v)+ε (in a smooth setting
0
one can take ε=0), (2.5) surely holds if one can prove that
d+ (cid:12) ∂ (cid:12)
A (cid:12) ≤−2λA , or its pointwise version (cid:12) As ≤−2λAs. (2.8)
dt t(cid:12)t=0 0 ∂t(cid:12)t=0 t 0
Having obtained the contraction property from (2.8), it still remains open how to deduce that F
is geodesically convex. Notice that along an arbitrary curve ηs
∂
∂sF(ηs)=(cid:104)∇F(ηs),∂sηs(cid:105)g =−(cid:104)∂rSr(ηs)|r=0,∂sηs(cid:105)g; (2.9)
applied to ηs :=γs, (2.9) and the semigroup property S (γs)=γs yield
t r t t+r
∂
F(γs)=−(cid:104)∂ γs,∂ γs(cid:105) . (2.10)
∂s t t t s t g
Inasmoothsetting wecanassumethatγsisaminimalgeodesic;operatingafurtherdi(cid:27)erentiation
with respect to s, we obtain
∂2 F(γs)(2=.9)− ∂ (cid:104)∂ γs,∂ γs(cid:105) (cid:12)(cid:12) =−(cid:104)D ∂ γs,∂ γs(cid:105) −(cid:104)∂ γs,D ∂ γs(cid:105) (cid:12)(cid:12) (2.11)
∂s2 ∂s t t s t g(cid:12)t=0 ∂s t t s t g t t ∂s s t g(cid:12)t=0
(cid:12) (cid:12) 1 ∂ (cid:12)
=−(cid:104)D ∂ γs,∂ γs(cid:105) (cid:12) =−(cid:104)D ∂ γs,∂ γs(cid:105) (cid:12) =− (cid:104)∂ γs,∂ γs(cid:105) (cid:12)
∂s t t s t g(cid:12)t=0 ∂t s t s t g(cid:12)t=0 2∂t s t s t g(cid:12)t=0
(2=.6)−21∂∂t(cid:12)(cid:12)(cid:12)t=0Ast (2≥.8)λ(cid:12)(cid:12)∂sγs(cid:12)(cid:12)2g, (2.12)
where we used the standard properties of the covariant di(cid:27)erentiations D ,D and, in (2.11),
∂s ∂t
the fact that at t=0 D ∂ γs =0, being γs =γs a geodesic.
∂s s t t
5
A metric derivation of convexity. Evenifthepreviousdi(cid:27)erentialargumentshowsthat(2.8)
implies geodesic λ-convexity, it still requires nice smooth properties on geodesics and covariant
di(cid:27)erentiation, which could be hard to extend to a non smooth setting.
This is not at all surprising, since the contraction property (2.5) and its action-di(cid:27)erential
characterization (2.8) do not carry all the information linking the semigroup S to F: in order to
conclude the argument in (2.11) we had therefore to insert the information coming from (2.9).
To overcome these di(cid:30)culties, we shall deal with a more precise metric characterization of S
than (2.4). As it has been proposed and studied in [4], gradient (cid:29)ows of geodesically λ-convex
functionals in (cid:16)almost(cid:17) Euclidean settings should satisfy a purely metric formulation in terms of
the Evolution Variational Inequality
1d+ λ
d2(S (u),v)+ d2(S (u),v)+F(S (u))≤F(v), ∀v ∈X, t>0. (2.13)
t t t
2 dt 2
It can be proved (see [5]) that (2.13) characterizes S and implies the contractivity property (2.4).
As we discussed before, here we invert the usual procedure (starting from a convex functional,
construct its gradient (cid:29)ow) and we suppose that there exists a smooth (cid:29)ow S satisfying (2.13).
t
The following result, whose proof will be postponed (in a more general form) to Theorem 3.2 in
the next Section, shows that F is geodesically λ-convex.
Theorem 2.1 Suppose that there exists a continuous semigroup of maps S ∈ C0(X;X), t ≥ 0,
t
satisfying (2.13). Then for every (minimal, constant speed) geodesic γ :[0,1]→X
λ
F(γs)≤(1−s)F(γ0)+sF(γ1)− s(1−s)d2(γ0,γ1), ∀s∈[0,1] (2.14)
2
i.e. F is (strongly) geodesically λ-convex.
E.V.I.throughaction-di(cid:27)erentialestimates. ThankstoTheorem2.1,itispossibletoprove
the geodesic λ-convexity of F by exhibiting a (cid:29)ow S satisfying the E.V.I. (2.13). According to
the general strategy suggested by [19], we want to reduce (2.13) to a suitable family of di(cid:27)erential
inequalities satis(cid:28)ed by the action As of (2.6).
t
The idea here is to consider a di(cid:27)erent family of perturbations of a given smooth curve γ :
[0,1] → X, still induced by the semigroup S. In fact, di(cid:27)erently from the contraction estimate
(2.5) where we are (cid:29)owing both the points u,v through S , in (2.13) we want to keep the point
t
v :=γ0 (cid:28)xedandtovaryonlyu:=γ1. Ifγs isasmoothcurveconnectingthem, itisthennatural
to consider the new families (see Figure 1)
γ˜s :=S (γs)=γs, F˜s :=F(γ˜s) s∈[0,1], t≥0. (2.15)
t st st t t
u =S (u)=γ1 =γ˜1
t t t t
γs
t
γ0
t
)
s
γ S
(
Sst
=
S s
γ˜t
u=γ1
v =γ0 γs
Figure 1: variation of the curve γs under the action of the semigroup S.
6
1
Notice that γ˜s =γs, γ˜0 =γ0 =v, γ˜1 =S (γ1)=S (u). As before, we introduce the quantities
0 t t t t
(cid:90) 1
A˜st :=(cid:12)(cid:12)∂sγ˜ts(cid:12)(cid:12)2g, A˜t := A˜stds. (2.16)
0
Theorem 2.2 (A di(cid:27)erential inequality linking action and (cid:29)ow) Supposethatforeverysmooth
curve γ :[0,1]→X the quantities A˜s,F˜s induced by the (cid:29)ow S through (2.15),(2.16) satisfy
t t
1 ∂ ∂
A˜s+ F˜s ≤−λsA˜s, ∀t≥0. (2.17)
2∂t t ∂s t t
Then S satis(cid:28)es (2.13), it is the gradient (cid:29)ow of F, and F is geodesically λ-convex. Moreover, it
is su(cid:30)cient to check (2.17) at t=0.
Proof. Let us (cid:28)rst observe that (2.17) yields, after an integration with respect to s in [0,1],
1 d (cid:90) 1
A˜+F˜1−F˜0 ≤−λ sA˜sds. (2.18)
2dt t t t t
0
By the semigroup property, it is su(cid:30)cient to prove (2.13) at t = 0. We choose a geodesic γs
connecting v to u and we consider the curves given by (2.15). Since
(cid:90) 1 (cid:90) 1
d2(v,S (u))≤ A˜sds=A˜, d2(v,u)= A˜sds=A˜, F˜1 =F(S (u)), F˜0 =F(v),
t t t 0 0 t t t
0 0
(2.19)
by (2.18) at t=0 we obtain
1d+ (cid:12) (cid:90) 1 λ
d2(S (u),v)(cid:12) +F(u)−F(v)≤−λ sA˜sds=− d2(u,v), (2.20)
2 dt t (cid:12)t=0 0 0 2
where in the last identity we used the fact that γs is a geodesic and therefore A˜s = |∂ γs|2 is
0 s g
constant in [0,1] and takes the value d2(γ0,γ1)=d2(v,u).
Since γ˜s =S γ˜s by the semigroup property, if S satis(cid:28)es (2.17) at the initial time t=0 for
an arbitrarty0+stmootsht ctu0rve γ, then it also satis(cid:28)es (2.17) for t>0. (cid:3)
Our last result provides a simple criterion to check (2.17):
Theorem 2.3 Suppose that the (cid:29)ow S : [0,+∞)×X → X satis(cid:28)es (2.9) for any smooth curve
γs, let γs,γ˜s,As,A˜s,F˜s be de(cid:28)ned as in (2.6), (2.15), and (2.16), and let us set
t t t t t
D˜rs := 12lhi↓m0h−1(cid:16)(cid:12)(cid:12)∂sγssr+h(cid:12)(cid:12)2g−(cid:12)(cid:12)∂sγssr(cid:12)(cid:12)2g(cid:17), (2.21)
Then
1 ∂ ∂
A˜s+ F˜s =sD˜s. (2.22)
2∂t t ∂s t t
Furthermore, if (2.8) holds, then
D˜s ≤−λA˜s (2.23)
t t
and (2.17) holds, too, so that F is geodesically λ-convex, and S is its gradient (cid:29)ow.
Proof. Let us set
γ˜ts,τ :=Sτγ˜ts =γsst+τ, A˜st,τ :=(cid:12)(cid:12)∂sγ˜ts,τ(cid:12)(cid:12)2g, (2.24)
so that
(cid:12) 1 ∂ (cid:12)
γ˜s =γ˜s , ∂ γ˜s =∂ γ˜s +h∂ γ˜s (cid:12) , D˜s = A˜s (cid:12) (2.25)
t+h t,sh s t+h s t,τ τ t,τ(cid:12)τ=sh t 2∂τ t,τ(cid:12)τ=0
7
Observe that the identity
|x+y|2 =2(cid:104)x+y,y(cid:105) +|x|2−|y|2, ∀x,y ∈T Mn (2.26)
g g g g γ
yields
A˜st+h =(cid:12)(cid:12)∂sγ˜ts+h(cid:12)(cid:12)2g (2=.25)(cid:12)(cid:12)∂sγ˜ts,τ +h∂τγ˜ts,τ(cid:12)(cid:12)2g(cid:12)(cid:12)(cid:12)τ=sh
(2=.26)(cid:104)2h(cid:104)∂sγ˜ts,τ +h∂τγ˜ts,τ,∂τγ˜ts,τ(cid:105)+(cid:12)(cid:12)∂sγ˜ts,τ(cid:12)(cid:12)2g−h2(cid:12)(cid:12)∂τγ˜ts,τ(cid:12)(cid:12)2g(cid:105)τ=sh
=2h(cid:104)∂ γ˜s ,∂ S (γ˜s ))(cid:105)(cid:12)(cid:12) +A˜s −o(h)(2=.9)−2h ∂ F(γ˜s )+A˜s −o(h).
s t+h θ θ t+h (cid:12)θ=0 t,sh ∂s t+h t,sh
We thus get
1 (cid:0)A˜s −A˜s(cid:1)+ ∂ F(γ˜s )= 1 (cid:0)A˜s −A˜s(cid:1)−o(1), (2.27)
2h t+h t ∂s t+h 2h t,sh t
so that, passing to the limit as h↓0 we get (2.22). (cid:3)
Remark 2.4 Notice that the remainder term o(1) in (2.27) is non-negative, so it can be simply
neglected, if one is just interested to the inequality (2.17).
3 Gradient (cid:29)ows and geodesic convexity in a metric setting
In this section we will brie(cid:29)y recall some basic de(cid:28)nitions and properties of gradient (cid:29)ows in a
metric setting and we will prove Theorem 2.1 in a slightly more general framework.
Let (X,d) be a metric space (not necessarily complete) and let F :X→(−∞,+∞] be a lower
(cid:8) (cid:9)
semicontinuous functional, whose proper domain D(F) := w ∈ X : F(w) < +∞ is dense in X
(otherwise we can always restrict all the next statements to the closure of D(F) in X). We also
assume that F is bounded from below, i.e. F :=inf F(u)>−∞.
inf u∈X
A C0-semigroup S in C0(X;X) is a family S , t≥0, of continuous maps in X such that
t
(cid:0) (cid:1)
S (u)=S S (u) , limS (u)=S (u)=u ∀u∈X, t,h≥0. (3.1)
t+h h t t 0
t↓0
Given a real number λ∈R, we say that S is the λ-(gradient) (cid:29)ow of F if it satis(cid:28)es
S (X)⊂D(F) for every t>0; (3.2a)
t
the map t(cid:55)→F(S (u)) is not increasing in (0,+∞); (3.2b)
t
1d+ λ
d2(S (u),v)+ d2(S (u),v)+F(S (u))≤F(v), ∀u∈X, v ∈D(F), t≥0. (3.2c)
t t t
2 dt 2
Clearly, if S is a λ-(cid:29)ow for F, then it is also a λ(cid:48)-(cid:29)ow for every λ(cid:48) ≤ λ. The next proposition
collects some useful properties of λ-(cid:29)ows.
Proposition 3.1 (Integral characterization of (cid:29)ows and contraction) A C0-semigroup S
satis(cid:28)es (3.2a,b,c) if and only if it satis(cid:28)es the following integrated form
eλ(t1−t0) 1 (cid:16) (cid:17)
d2(S (u),v)− d2(S (u),v)≤E (t −t ) F(v)−F(S (u)) ∀0≤t <t , (3.3)
2 t1 2 t0 λ 1 0 t1 0 1
(cid:40)
for every u∈X, v ∈D(F), where E (t):=(cid:82)teλrdr = eλtλ−1 if λ(cid:54)=0,
λ 0 t if λ=0.
In particular S satis(cid:28)es the uniform regularization bound
1
F(S (u))≤F(v)+ d2(u,v) ∀u∈X, v ∈D(F), t>0, (3.4)
t
2E (t)
λ
8
the uniform continuity estimate
(cid:16) (cid:17)
d2(S (u),S (u))≤2E (t −t ) F(S u)−F ∀u∈D(F), 0≤t ≤t , (3.5)
t1 t0 −λ 1 0 t0 inf 0 1
and the λ-contraction property, i.e.
d(S (u),S (v))≤e−λtd(u,v) ∀u,v ∈X, t≥0. (3.6)
t t
Proof. Clearly (3.3) yields (3.2a), being D(F)(cid:54)=∅; (3.2b) and (3.5) follow by taking v :=S (u)
t0
and (3.2c) can be proved by dividing both sides of (3.3) by t −t and passing to the limit as
1 0
t ↓ t . In order to prove the converse implication, let us (cid:28)rst observe that for a continuous real
1 0
function ζ :[0,+∞)→R
ζ(t+h)−ζ(t)
liminf ≤0 ∀t>0 =⇒ ζ is not increasing. (3.7)
h↓0 h
In fact, if 0 ≤ t < t +τ existed with δ := τ−1(cid:0)ζ(t +τ)−ζ(t )(cid:1) > 0, then a minimum point
0 0 0 0
t¯∈[t ,t +τ) of t(cid:55)→ζ(t)−ζ(t )−δ(t−t ) would satisfy
0 0 0 0
ζ(t¯+h)−ζ(t¯)
liminf −δ ≥0, which contradicts (3.7).
h↓0 h
(3.3) then follows by (3.2c), after a multiplication by eλt and choosing
ζ(t):= eλtd2(S (u),v)+(cid:90) teλr(cid:0)F(S (u))−F(v)(cid:1)dr, t¯>0,
t r
2
t¯
and recalling the monotonicity property (3.2b). A similar argument shows that
1 1 λ(cid:90) t1 (cid:16) (cid:17)
d2(S (u),v)− d2(S (u),v)+ d2(S (u),v)dr ≤(t −t ) F(v)−F(S (u)) , (3.8)
2 t1 2 t0 2 r 1 0 t1
t0
for every 0 ≤ t < t , u ∈ X, and v ∈ D(F). In order to prove the λ-contracting property, we
0 1
apply (3.8) obtaining
d2(S (u),S (v))−d2(u,v)=d2(S (u),S (v))−d2(S (u),v)+d2(S (u),v)−d2(u,v)
h h h h h h
(cid:90) h(cid:16) (cid:17) (cid:16) (cid:17)
≤−λ d2(S (u),S (v))+d2(S (u),v) dr+2h F(v)−F(S (v)) .
h r r h
0
We divide this inequality by h and we pass to the limit as h ↓ 0; the continuity of S , the lower
t
semicontinuity of F, and the semigroup property of S yield
d+
d2(S (u),S (v))≤−2λd2(u,v) ∀u,v ∈X, t>0, (3.9)
t t
dt
which yields (3.6) thanks to (3.7). (cid:3)
We can now prove the main result of this section: if a functional F admits a λ-(cid:29)ow, then F is
geodesically λ-convex.
Theorem 3.2 (Geodesic convexity via E.V.I.) LetussupposethatSisaλ-(cid:29)owforthefunc-
tional F, according to (3.2a,b,c), and let γ :[0,1]→X be a Lipschitz curve satisfying
d(γr,γs)≤L|r−s|, L2 ≤d2(γ0,γ1)+ε2 ∀r,s∈[0,1], (3.10)
for some constant ε≥0. Then for every t>0 and s∈[0,1]
λ ε2
F(S (γs))≤(1−s)F(γ0)+sF(γ1)− s(1−s)d2(γ0,γ1)+ s(1−s). (3.11)
t
2 2E (t)
λ
9
In particular, when γ is a geodesic (i.e. γ satis(cid:28)es (3.10) with L=d(γ0,γ1), ε=0), we have
λ
F(γs)≤(1−s)F(γ0)+sF(γ1)− s(1−s)d2(γ0,γ1), (3.12)
2
i.e. F is (strongly) geodesically λ-convex.
Proof. Let γ be satisfying (3.10) and let us set γs := S (γs). Choosing t = 0, t = t, u := γs,
t t 0 1
and taking a convex combination of (3.3) written for v :=γ0, and v :=γ1, we get
eλt(cid:16) (cid:17) 1(cid:16) (cid:17)
(1−s)d2(γs,γ0)+sd2(γs,γ1) − (1−s)d2(γs,γ0)+sd2(γs,γ1) (3.13)
2 t t 2
(cid:16) (cid:17)
≤E (t) (1−s)F(γ0)+sF(γ1)−F(γs) . (3.14)
λ t
We now observe that the elementary inequality
(1−s)a2+sb2 ≥s(1−s)(a+b)2 ∀a,b∈R, s∈[0,1], (3.15)
and the triangular inequality yield
(3.15) (cid:16) (cid:17)2
(1−s)d2(γs,γ0)+sd2(γs,γ1) ≥ s(1−s) d(γs,γ0)+d(γs,γ1) ≥s(1−s)d(γ0,γ1)2. (3.16)
t t t t
On the other hand, (3.10) yields
(1−s)d2(γs,γ0)+sd2(γs,γ1)≤L2s(1−s). (3.17)
Inserting (3.17) and (3.16) in (3.14) we obtain
eλt−1 ε2 (cid:16) (cid:17)
s(1−s)d2(γ0,γ1)− s(1−s)≤E (t) (1−s)F(γ0)+sF(γ1)−F(γs) . (3.18)
2 2 λ t
Dividing then both sides of (3.18) by E (t) we get (3.11); when ε=0 we can pass to the limit as
λ
t↓0 obtaining (3.12). (cid:3)
Weconcludethissectionbyconsideringthecasewhenthe(cid:29)owSisonlyde(cid:28)nedonadense subset
X of D(F). In order to prove the geodesic convexity of F in X by Theorem 3.2 we (cid:28)rst have to
0
extend S to the whole space X. This can be achieved by a density argument, if X is complete and
the lower semicontinuous functional F satis(cid:28)es the following approximation property:
∀u∈X ∃u ∈X : lim d(u ,u)=0, lim F(u )=F(u). (3.19)
n 0 n n
n→∞ n→∞
We state the precise extension result in the next theorem.
Theorem 3.3 Suppose that the functional F and the subset X ⊂ D(F) satisfy (3.19) and let
0
S be a λ-(cid:29)ow for F in X . If X is complete, S can be extended to a unique λ-(cid:29)ow S¯ in X and
0
therefore F is (strongly) geodesically λ-convex in X.
Proof. Given u∈X and a sequence u ∈X as in (3.19), we can de(cid:28)ne
n 0
S¯ (u):= lim S (u ) ∀t>0, (3.20)
t t n
n→∞
where it is clear that the limit in (3.20) exists (being X complete and S Lipschitz by (3.6))
t
and does not depend on the particular sequence u we used to approximate u. Moreover S¯ is a
n t
semigroupandsatis(cid:28)estheestimate(3.5)andtheλ-contractingproperty (3.6); beingD(F)dense
in X, it is not di(cid:30)cult to combine (3.5), (3.6) and (3.19) to show that lim S (u) = u for every
t↓0 t
u∈X.
InordertoprovethatS¯ isstillaλ-(cid:29)owforF inXwehavetocheck(3.3)inX: we(cid:28)xv ∈D(F)
and a sequence v ∈ X converging to v with F(v ) → F(v) and we pass to the limit as s → ∞
n 0 n
in the inequalities
eλ(t1−t0)d2(S (u ),v )− 1d2(S (u ),v )≤E (t −t )(cid:0)F(v )−F(S (u )), (3.21)
2 t1 n n 2 t0 n n λ 1 0 n t1 n
using the lower semicontinuity of F. (cid:3)
10