First variation formula in Wasserstein spaces over compact Alexandrov spaces

Transcription

1 Fir variaion formula in Waerein pace over compac Alexandrov pace Nicola Gigli and Shin-ichi Oha July 29, 2010 Abrac We exend reul proven by he econd auhor Amer. J. ah., 2009 for nonnegaively curved Alexandrov pace o general compac Alexandrov pace X wih curvaure bounded below: he gradien flow of a geodeically convex funcional on he quadraic Waerein pace PX, W 2 aifie he evoluion variaional inequaliy. oreover, he gradien flow enjoy uniquene and conraciviy. Thee reul are obained by proving a fir variaion formula for he Waerein diance. 1 Inroducion Thi paper hould be conidered a an addendum o [Oh] of he econd auhor. In [Oh], i i udied he quadraic Waerein pace PX, W 2 buil over a compac Alexandrov pace X wih curvaure bounded below, and proven he exience of Euclidean angen cone ee alo [Gi]. Thi reul i paricularly inereing for Alexandrov pace wih a negaive curvaure bound, a i i known ha Waerein pace buil over hem do no admi lower curvaure bound in he ene of Alexandrov. The exience of uch angen cone ha hen been ued in [Oh] o perform udie of gradien flow of geodeically convex funcional on PX, W 2. In paricular, exience of gradien flow of uch funcional ha been proven via an approach inpired by [Ly] which differ from he purely meric approach in [AGS] wihou uing angen cone. For echnical reaon, however, uniquene and conracion of uch gradien flow have been obained only in he cae where X ha he nonnegaive curvaure. In hi paper we exend hee laer reul o general Alexandrov pace wih curvaure bounded below poibly by a negaive value Theorem 4.2. A key ool in our approach i a fir variaion formula for he Waerein diance Theorem 3.4 from which i alo eaily follow ha gradien flow aify he evoluion variaional inequaliy Propoiion 4.1. See Remark 3.7, 4.5 for he difference from he argumen in [Oh]. Secion 2 i devoed o recalling known reul on he geomeric rucure of and gradien flow in PX, W 2. We how he fir variaion formula in Secion 3, and ue i o udy gradien flow in Secion 4. Iniu für Angewande ahemaik, Univeriä Bonn, Endenicher Allee 60, Bonn, Germany nicolagigli@googl .com Deparmen of ahemaic, Kyoo Univeriy, Kyoo , Japan oha@mah.kyoo-u.ac.jp; Suppored in par by he Gran-in-Aid for Young Scieni B

2 2 Preliminarie 2.1 Waerein pace over compac Alexandrov pace Le X, d be a meric pace. A recifiable curve γ : [0, l] X i called a geodeic if i i locally minimizing and ha a conan peed. We ay ha γ i minimal if i i globally minimizing i.e., dγ, γ = /l dγ0, γl for all, [0, l]. If any wo poin in X are joined by ome minimal geodeic, hen X i called a geodeic pace. Throughou he aricle, X, d will be a compac Alexandrov pace of curvaure 1. By hi we mean ha X, d i a geodeic pace uch ha every riangle in X i hicker han a geodeic riangle wih he ame ide lengh in he hyperbolic plane H 2 1 of conan ecional curvaure 1 ee [Oh] for he deailed definiion, [BGP], [OSh] and [BBI] for he baic heory. We remark ha X, d can be infinie-dimenional, o ha i local rucure may be very wild. Denoe by PX he e of all Borel probabiliy meaure on X. Given µ, ν PX, we conider he L 2 -Waerein diance { 1/2 W 2 µ, ν := inf dx, y 2 dπx, y}, π X X where π PX X run over all coupling of µ and ν. Noe ha W 2 µ, ν i finie and PX, W 2 i compac a X i aumed o be compac. We refer o [AGS] and [Vi] for more on Waerein geomery and opimal ranpor heory. I i known ha, if X, d ha nonnegaive curvaure, hen o doe PX, W 2 [S2, Propoiion 2.10], [LV, Theorem A.8]. Alhough he analogue implicaion i fale for negaive curvaure bound eenially becaue i i no a caling invarian condiion, [S2, Propoiion 2.10], we obain he following generalizaion of he 2-uniform moohne in Banach pace heory. Propoiion 2.1 [Oh, Propoiion 3.1, Lemma 3.3], [Sa] For all µ, ν, ω PX, all minimal geodeic α : [0, 1] PX from ν o ω and for all τ [0, 1], we have W 2 µ, ατ 2 1 τw2 µ, ν 2 + τw 2 µ, ω 2 S 2 1 ττw 2 ν, ω 2, 1 where S = 1 + diam X 2. In fac, he 2-uniform moohne 1 hold in X, d and decend o PX, W 2 wih he ame conan S. Alhough PX, W 2 i no an Alexandrov pace, i i poible o how he following ee alo [Oh, Theorem 3.6]: Theorem 2.2 [Gi, Theorem 3.4, Remark 3.5] Given µ PX and uni peed geodeic α, β : [0, δ PX wih α0 = β0 = µ, he join limi exi W 2 α, β 2 lim, 2 2

3 Theorem 2.2 mean ha PX, W 2 look like a Riemannian pace raher han a Finler pace, and we can inveigae i infinieimal rucure according o he heory of Alexandrov pace. For µ PX, denoe by Σ µ[px] he e of all nonrivial uni peed minimal geodeic emanaing from µ. Given α, β Σ µ[px], Theorem 2.2 verifie ha he angle µ α, β := arcco lim, W 2 α, β 2 [0, π] 2 i well-defined and provide an appropriae peudo-diance rucure of Σ µ[px]. We define he pace of direcion Σ µ [PX], µ a he compleion of Σ µ[px]/, µ, where α β hold if µ α, β = 0. Then he angen cone C µ [PX], σ µ i defined a he Euclidean cone over Σ µ [PX], µ : / C µ [PX] := Σ µ [PX] [0, Σ µ [PX] {0}, σ µ α,, β, := co µ α, β. Noe ha σ µ i a diance on C µ [PX] and ha σ µ α,, β, = lim ε 0 W 2 αε, βε /ε 2 hold by he definiion of µ. We denoe he origin of C µ [PX] by o µ and define α,, β, µ := 2 co µ α, β, α, µ := = σ µ oµ, α, for α,, β, C µ [PX]. The ubcrip µ will be omied if he pace under conideraion i clearly underood. We omeime abbreviae like α, := α, and idenify α Σ µ [PX] wih α, 1 C µ [PX]. Uing hi infinieimal rucure, we inroduce a cla of differeniable curve. Definiion 2.3 Righ differeniabiliy We ay ha a curve ξ : [0, l PX i righ differeniable a [0, l if here i v C ξ [PX] uch ha, for any equence {ε i } i N of poiive number ending o zero and {α i } i N of uni peed minimal geodeic from ξ o ξ + ε i, he equence {α i, W 2 ξ, ξ + ε i /ε i } i N C ξ [PX] converge o v. Such v i clearly unique if i exi, and hen we wrie ξ = v. In paricular, we have lim ε 0 W 2 ξ, ξ + ε/ε = ξ ξ. We alo remark ha every minimal geodeic α : [0, l] PX i righ differeniable a all [0, l. Thi i becaue Alexandrov pace are known o aify he non-branching propery, which i inheried by PX, W 2 cf. [Vi, Corollary 7.32]. Therefore α [0,] i a unique minimal geodeic beween α0 and α for all 0, l, and α0 = α, W 2 α0, α/. 2.2 Gradien flow in Waerein pace The conen of hi ubecion will come ino play in Secion 4. The reader inereed only in he fir variaion formula can kip o Secion 3. 3

4 Conider a lower emi-coninuou funcion f : PX, + ] which i K-convex for ome K R in he ene ha f ατ 1 τf α0 + τf α1 K 2 1 ττw 2 α0, α1 2 3 hold along all minimal geodeic α : [0, 1] PX and all τ [0, 1]. We alo uppoe ha f i no idenically +, and define P X := {µ PX fµ < }. The K-convexiy guaranee ha minimal geodeic beween poin in P X are again included in P X, and hence i make ene o conider Σ µ [P X] a well a C µ [P X] for µ P X. Given µ P X and α Σ µ [P X], we e D µ fα := lim inf β α lim fβ fµ, where β Σ µ[p X] i a uni peed geodeic and he convergence β α i wih repec o µ. Clearly D µ f i well-defined on Σ µ [P X]. Define he abolue gradien called he local lope in [AGS] of f a µ P X by f µ := max { 0, lim up ν µ } fµ fν. W 2 µ, ν Noe ha D µ fα f µ hold for any α Σ µ [P X]. According o he argumen in [PP] and [Ly], we find a negaive gradien vecor of f a each poin in P X wih finie abolue gradien. Lemma 2.4 [Oh, Lemma 4.2] For each µ P X wih 0 < f µ <, here exi unique α Σ µ [P X] aifying D µ fα = f µ. oreover, for any β Σ µ [P X], i hold ha D µ fβ f µ α, β µ. The econd aerion i regarded a a fir variaion formula for f. Uing α in he above lemma, we define he negaive gradien vecor of f a µ a fµ := α, f µ C µ [P X]. In he cae of f µ = 0, we imply e fµ := o µ. Definiion 2.5 Gradien curve A coninuou curve ξ : [0, l P X which i locally Lipchiz coninuou on 0, l i called a gradien curve of f if f ξ < for all 0, and if i i righ differeniable wih ξ = fξ a all [0, l wih f ξ <. We ay ha a gradien curve ξ i complee if i i defined on [0,. Again along he dicuion in [PP] and [Ly], depie ome echnical difficulie a PX, W 2 i no an Alexandrov pace, we can how he exience of complee gradien curve. 4

5 Theorem 2.6 [Oh, Theorem 5.11] For any µ P X, here exi a complee gradien curve ξ : [0, P X of f wih ξ0 = µ. Remark 2.7 Le u make a more deailed commen on he conrucion of gradien curve. The raegy in [Oh] following [PP] and [Ly] i ha we fir conruc a uni peed curve η wih f η = f η a.e., hen an appropriae reparamerizaion of η provide a gradien curve. Anoher way i he direc conrucion comprehenively dicued in [AGS]. In fac, a generalized minimizing movemen u : [0, PX i locally Lipchiz coninuou on 0, and aifie fu + ε fu lim ε 0 ε = f u { } 2 2 W 2 u, u + ε = lim ε 0 ε a every 0, [AGS, Theorem ], and herefore he dicuion a in [Oh, Lemma 5.5] how ha u i a gradien curve in he ene of Definiion 2.5. oreover, he uniquene of gradien curve proved below Theorem 4.2 enure ha boh conrucion give rie o he ame curve. 3 A fir variaion formula Thi ecion conain our main reul. Thee are hown afer a erie of lemma. Lemma 3.1 For any minimal geodeic α : [0, ] PX and ν PX, we have W 2 α, ν 2 W 2 α0, ν 2 lim inf 0 W 2 α, ν 2 W 2 α0, ν 2 + S2 W 2, 2 α0, α where S = 1 + diam X 2. Proof. For 0,, he 2-uniform moohne 1 how 2 W 2 α, ν 1 W 2 α0, ν 2 + W 2 Dividing boh ide by yield 2 α, ν S 2 1 W 2. 2 α0, α W 2 α, ν 2 W 2 α0, ν 2 W 2α, ν 2 W 2 α0, ν 2 + S2 1 2, W 2 α0, α and leing end o zero complee he proof. 5

6 Lemma 3.2 For any pair of minimal geodeic α : [0, δ PX, β : [0, 1] PX wih α0 = β0 =: µ, we have Proof. lim up 0 W 2 α, β1 2 W 2 µ, β1 2 For any 0, δ and 0, 1, he riangle inequaliy give W 2 α, β1 W 2 µ, β1 Thu we deduce from 2 ha, for any > 0, lim up 0 Leing go o infiniy implie W 2 α, β1 W 2 µ, β1 lim up 0 2 α0, β0 µ. 4 W 2α, β + W 2 β, β1 W 2 µ, β1 = W 2α, β W 2 µ, β. W 2 α, β1 2 W 2 µ, β1 2 = 2W 2 µ, β1 lim up 0 σ α0, β0 β0 = α 2 2 α β co α, β σ α, β + β 0. W 2 α, β1 W 2 µ, β1 2 β 2 α β co α, β 2 β 0 = 2 α0, β0. We remark ha equaliy hold in 4 if X i nonnegaively curved cf. [BBI, Theorem 4.5.6]. Lemma 3.3 For any riple v 1, v 2, w C µ [PX], we have v 1, w µ v 2, w µ + w µ σ µ v 1, v 2. Proof. We ju calculae, puing v i = α i, i and w = β,, v 1, w v 2, w 2 = 2{ 1 co α 1, β 2 co α 2, β } 2 = [ 2 2 1{1 in 2 α 1, β} + 2 2{1 in 2 α 2, β} ] co α 1, β co α 2, β [ { co α1, β co α 2, β + in α 1, β in α 2, β }] { = co α 1, β α 2, β } 2{ co α 1, α 2 } = 2 σv 1, v

7 Now we are ready o prove our main heorem. Theorem 3.4 Fir variaion formula Le ξ : [0, δ PX be a curve righ differeniable a 0, and le β : [0, 1] PX be a minimal geodeic from µ := ξ0 o ν. Then we have lim up W 2 ξ, ν 2 W 2 µ, ν 2 2 ξ0, β0 µ. Proof. For each mall > 0, le α : [0, ] PX be a minimal geodeic from µ o ξ. Then i follow from he righ differeniabiliy of ξ ha α 0 converge o ξ0 in C µ [PX]. We deduce from Lemma 3.1, 3.2, 3.3 ha W 2 ξ, ν 2 W 2 µ, ν 2 = W 2α, ν 2 W 2 µ, ν 2 2 α 0, β0 + S2 W 2 µ, α 2 2 ξ0, β0 + 2 β0 σ ξ0, α 0 + S2 W 2 Leing end o zero, we complee he proof. µ, ξ 2. The following imple lemma valid for general meric pace i ueful. Lemma 3.5 Le ξ, ζ : [0, δ Y be curve in a meric pace Y, d, and z Y be a midpoin of x := ξ0 and y := ζ0 i.e., dz, x = dz, y = dx, y/2. Then we have lim up 2 lim up dξ, ζ 2 dx, y 2 dξ, z 2 dx, z lim up dζ, z 2 dy, z 2. Proof. The riangle inequaliy immediaely implie dξ, ζ 2 dx, y 2 dξ, ζ dx, y lim up = 2dx, y lim up dξ, z + dζ, z dx, y 2dx, y lim up { } dξ, z dx, z dζ, z dy, z 2dx, y lim up + lim up dξ, z 2 dx, z 2 dζ, z 2 dy, z 2 = 2 lim up + 2 lim up. Combining hi wih Theorem 3.4 yield he following fir variaion formula for he diance beween wo righ differeniable curve. 7

8 Corollary 3.6 Le ξ, ζ : [0, δ PX be wo curve righ differeniable a 0. Pu µ := ξ0, ν := ζ0, le α : [0, 1] PX be a minimal geodeic from µ o ν and le βτ := α1 τ be i convere curve. Then we have lim up W 2 ξ, ζ 2 W 2 µ, ν 2 2 ξ0, α0 µ 2 ζ0, β0 ν. Proof. Apply Lemma 3.5 o ξ and ζ wih z = α1/2 = β1/2. Then Theorem 3.4 yield he deired eimae. Remark 3.7 If X i nonnegaively curved, hen PX, W 2 i an Alexandrov pace and hence he righ differeniabiliy give a beer conrol a he level of PX no only in C µ [PX]. Such rong righ differeniabiliy [Oh, 6.1] and he fir variaion formula along geodeic Lemma 3.2 immediaely lead u o he formula along righ differeniable curve ee [Oh, Lemma 6.1]. 4 Applicaion for gradien flow in PX In hi ecion, we ue he fir variaion formula in he previou ecion o exend reul in [Oh, Secion 6] where we aumed ha X, d i nonnegaively curved. 4.1 Uniquene and conracion A in Subecion 2.2, le f : PX, + ] be a lower emi-coninuou, K-convex funcion for ome K R uch ha P X = f 1, + i nonempy. We fir verify he evoluion variaional inequaliy ee [AGS, ] a a conequence of fir variaion formula Lemma 2.4 and Theorem 3.4. Propoiion 4.1 Evoluion variaional inequaliy Le ξ : [0, P X be a gradien curve of f. Then we have, for any 0, and ν PX, lim up ε 0 W 2 ξ + ε, ν 2 W 2 ξ, ν 2 2ε + K 2 W 2 ξ, ν 2 + f ξ fν. Proof. The aerion i clear if fν =, o ha we aume ν P X. We oberve from Theorem 3.4 ha lim up ε 0 W 2 ξ + ε, ν 2 W 2 ξ, ν 2 2ε ξ, β0 ξ, where β : [0, 1] P X i a minimal geodeic from ξ o ν. A ξ = fξ, Lemma 2.4 and he K-convexiy 3 of f ogeher imply ξ, β0 ξ lim τ 0 fβτ fξ τ Thi complee he proof. fν f ξ K 2 W 2 ξ, ν 2. A imilar argumen how he conracion propery of gradien curve. 8

9 Theorem 4.2 Conracion and uniquene Given any pair of gradien curve ξ, ζ : [0, P X of f, W 2 ξ, ζ e K W 2 ξ0, ζ0 hold for all [0,. In paricular, each µ P X admi a unique complee gradien curve aring from µ. Proof. Pu h := W 2 ξ, ζ 2, fix 0,, le α : [0, 1] PX be a minimal geodeic from ξ o ζ and pu βτ := α1 τ. Then Corollary 3.6 and Lemma 2.4 how ha lim up ε 0 { 2 h + ε h ε lim τ 0 fατ fα0 τ 2 ξ, α0 ξ 2 ζ, β0 ζ } fβτ fβ0 + lim 2Kh. τ 0 τ We ued he K-convexiy 3 along α in he la inequaliy. Therefore we obain h 2Kh for a.e., and hence h e 2K h0 by Gronwall heorem. We define he gradien flow G : P X [0, P X a ξ := Gµ, i he unique gradien curve aring from ξ0 = µ. Noe ha G i coninuou by virue of he conracion propery. Corollary 4.3 The gradien flow G : P X [0, P X exend uniquely and coninuouly o G : P X [0, P X. oreover, G aifie he conracion propery: W 2 Gµ,, Gν, e K W 2 Gµ, 0, Gν, 0 for µ, ν P X and 0, ; a well a he emigroup propery: Gµ, + = G Gµ,, for µ P X and, [0,. 4.2 Hea flow a gradien flow on Riemannian manifold Unil here, we only deal wih he riangle comparion propery of he Waerein pace. In hi la ubecion, in order o ee ha gradien flow of he free energy produce a oluion o he Fokker-Planck equaion, we ue he rucure of he underlying pace ha wa implicily avoided in [Oh]. Thi kind of inerpreaion of evoluion equaion goe back o celebraed work of Jordan e al. [JKO]. I i recenly demonraed ha here i alo a remarkable connecion wih he Ricci flow ee [T]. Alhough ome par alo work in Alexandrov pace, we conider only compac Riemannian manifold for breviy he compacne guaranee ha he ecional curvaure i bounded below. Since Theorem 4.2 enure ha our gradien curve coincide wih he one conruced a in [AGS] ee Remark 2.7, he realizaion of oluion o he Fokker-Planck equaion a gradien flow of he free energy i a well eablihed fac in 9

10 he Riemannian eing. Here, however, we preen a way of compleing he elf-conained proof in [Oh, Subecion 6.2] along our noion of gradien curve for horoughne. Le, g be a compac Riemannian manifold equipped wih he aociaed Riemannian diance d and he volume meaure vol g. Thank o ccann heorem [c], we can repreen each v C µ [P] a a meaurable vecor field on which will be again denoed by v by a ligh abue of noaion. oreover, for v, w C µ [P], we have σ µ v, w 2 = vx wx 2 dµx. In oher word, Oo [O] Riemannian rucure coincide wih our induced from Theorem 2.2. Due o he compacne of, he Taylor expanion immediaely give he following: Lemma 4.4 For any h C and any geodeic α : [0, l P, we have h dµ = h dµ + v, h dµ + O h W2 µ, µ 2, where we e µ := α0, µ := α and v := α0 C µ [P]. Le f : P, + ] be a in Subecion 2.2, 4.1. Take µ P and le ξ : [0, P be he unique complee gradien curve wih ξ0 = µ. Fix > 0 and recall ha ξ i righ differeniable wih ξ = fξ. For any h C, ince he remainder erm O h in Lemma 4.4 i uniform in he choice of geodeic α, we obain { } 1 lim h dµ +δ h dµ = fµ, h dµ, δ 0 δ where we e µ := ξ. For each δ > 0, chooe ome ν δ P aaining he infimum of he funcion P ν fν + W 2µ, ν 2. 2δ Such ν δ indeed exi ince P i compac and f i lower emi-coninuou. We alo chooe a minimal geodeic β δ : [0, l δ ] P from µ o ν δ where l δ := W 2 µ, ν δ. Then we know ha β δ, l δ /δ C µ [P] converge o fµ a δ end o zero [Oh, Lemma 6.4]. Thu Lemma 4.4 alo how ha, for any h C, { } 1 lim h dν δ h dµ = fµ, h dµ. δ 0 δ Therefore we conclude ha { 1 lim δ 0 δ hold for all h C. h dµ +δ h dν δ } = 0 5 Remark 4.5 If X ha he nonnegaive curvaure, hen he convergence of β δ, l δ /δ o fµ implie lim δ 0 W 2 ν δ, µ +δ /δ = 0 ee [Oh, Lemma 6.4]. Thu he Kanorovich- Rubinein heorem yield 5 wihou uing Lemma 4.4, moreover, he convergence 5 i uniform for all 1-Lipchiz funcion h. 10

11 For µ P, we define he relaive enropy a { Enµ := ρ log ρ dvol g if µ = ρ vol g, + oherwie. Given V C, le f V : P, ] be he aociaed free energy: fµ := Enµ + V dµ. Noe ha f V i lower emi-coninuou and he correponding ube P P aifie P = P. Furhermore, he K-convexiy of f V i known o be equivalen o he lower bound of he Bakry-Émery enor: Ric + He V K [S1] in paricular, he K-convexiy of En i equivalen o Ric K, [vrs]. Hence f V i K-convex for ome K R by virue of he compacne of. The eimae 5 i enough o follow he proof of [Oh, Theorem 6.6] and yield he following: Theorem 4.6 Given V C, a gradien curve ξ = ρ vol g : [0, P of f V produce a weak oluion ρ o he aociaed Fokker-Planck equaion: ρ = ρ + divρ V. In paricular, he gradien flow of En coincide wih he hea flow. To be precie, for any h C R and 0 1 < 2 <, we have 2 { } h h 2 dµ 2 h 1 dµ 1 = + h h, V dµ d, where we e µ := ξ and h := h,. 1 Remark 4.7 A hi poin, i hould be recalled ha in he Riemannian eing here are wo way o ee he hea flow a gradien flow: a gradien flow of he relaive enropy wih repec o he diance W 2 a we ju did, or a gradien flow of he Dirichle energy wih repec o he L 2 -diance aociaed wih he volume meaure. Recenly he econd auhor and Surm [OhS] proved ha hee wo approache coincide alo in he Finler eing. For finie-dimenional Alexandrov pace, he conrucion of he hea kernel via he Dirichle energy ha been performed in [KS]. I i ye o be proven ha uch hea kernel coincide wih he gradien flow of he enropy wih repec o W 2 in he genuine Alexandrov eing. Reference [AGS] L. Ambroio, N. Gigli and G. Savaré, Gradien flow in meric pace and in he pace of probabiliy meaure, Birkhäuer Verlag, Bael,

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