Spectral gaps for the O-U/Stochastic heat processes on path space over a Riemannian manifold with boundary

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1 arxiv: v1 [mah.pr] 5 Dec 18 ec1 Specral gap for he O-U/Sochaic hea procee on pah pace over a Riemannian manifold wih boundary Bo Wu School of Mahemaical Science, Fudan Univeriy, Shanghai 433, China wubo@fudan.edu.cn Abrac Fang-Wu[15] preened a explici pecral gap for he O-U proce on pah pace over a Riemannian manifold wihou boundary under he bounded Ricci curvaure condiion. In hi paper, we will exend hee reul o he cae of he Riemannian manifold wih boundary. Moreover, we alo derive he imilar reul for he ochaic hea proce. Keyword: Funcional inequaliy; Ricci Curvaure; Second fundamenal form; Diffuion proce; Pah pace. 1 Inroducion Funcional inequaliy i an imporan ool o udy he pecral gap for ome diffuion operaor in he analyi/ochaic analyi field, epecially, for he cae of infinie dimenional Riemannian pah pace. For he manifold wihou boundary, Fang[1] fir eablihed he Poincaré inequaliy for he O-U operaor on Riemanian pah pace by he Clark-Ocean formula, afer ha he log-sobolev inequaliy/weakpoincaré inequaliy have alo been eablihed for he O-U Dirichle form, ee e.g. [1, 3, 9, 18, 19, 3, 4, 1, 7, 8, 5] and reference herein. Recenly Naber[1] gave ome characerizaion of he uniform bound of Ricci curvaure by he analyi of he pah pace. Moiviaed by hi work, Fang and Wu[15] gave he explici pecral gap of he O-U operaor on Suppored in par by NNSFC

2 pah pace under he Ricci curvaure condiion ha K Ric Z : Ric+ Z K 1 and K +K 1. Thi condiion K +K 1 i removed by Cheng-Thalimaier[5]. For he manifold wih boundary, Wang[6] proved he damped log-sobolev inequaliy for he O-U proce on pah pace, bu ome geomeric informaion are hidden in hi inequaliy. In hi aricle, our main aim i o preen a eimae of he pecral gap for he O-U operaor on pah pace over a manifold poible wih boundary under he curvaure and he econd fundamenal form condiion eq K Ric Z K 1, σ I σ 1 foromeconank,k 1,σ,σ 1 R. Inparicular, ourreulcover Fang-Wu reul and Cheng-Thalmaier reul. Moreover, we alo obain he eimae of he pecral gap for he ochaic hea proce. To ae our main reul, we need o inroduce ome noaion. Le M be a d- dimenional complee Riemannian manifold poibly wih a boundary M and N be he inward uni normal vecor field of M. Le L 1 +Z be he diffuion operaor for ome C 1 vecor field Z, where i he Laplace operaor on M. Denoe by he Riemannian pah pace: W T x M {γ C[,T];M : γ x}. Le ρ be he Riemannian diance on M. Then Wx T M i a Polih pace under he uniform diance ρ γ,σ : up ργ,σ, γ,σ Wx T M. [,T] Le OM be he orhonormal frame bundle over M and π : OM M be he canonical projecion. Furhermore, we chooe a canonical orhonormal bai {e i } 1 i d on R d and a andard orhonormal bai H i u : H uei 1 i d for u OM of horizonal vecor field on OM. Then he horizonal reflecing diffuion proce i he unique oluion o he SDE: eq1. 1. du x H i U x dw +H Z U x d+h NU x dlx, Ux O xm, where W i he d-dimenional Brwonian moion on a complee filraion probabiliy pace Ω,{F },P, H Z and H N are he horizonal lif of Z and N repecively, and l x i an adaped increaing proce which increae only when X x : πu x M which i called he local ime of X x on M. Then i i eay o know ha X x olve he equaion eq dx x U x dw +ZX x d+nx dl x, Xx x up o he life ime ζ he maximal ime of he oluion.

3 Le FCT be he pace of bounded Lipchiz coninuou cylinder funcion on Wx TM, i.e. for every F FC T, here exi ome N 1 and < 1 < < N T, f C Lip M N uch ha Fγ fγ 1,,γ N,γ Wx T M, where C Lip M N i he collecion of bounded Lipchiz coninuou funcion on M N. Suppoe H i he andard Cameron-Marin pace for C[,T];R d, i.e. { } H h C[,T];R d : h, h H T : h d <. In order o conruc O-U proce on pah pace by he heory of Dirichle form, we fir inroduce he damped Mallavin gradien given by Wang[5]. To do ha, we will recommend a muliplicaive funcional Q x,, which i fir inroduced by Hu [] o inveigae gradien eimae on P. For any fixed, Q x, i an adaped righ-coninuou proce on R d R d uch ha Q x, P U x if X x M and eq Q x, I Q x,r{ RicZ U x r d+iux r dlx r where P u : R d R d i he projecion along u 1 N, i.e. } I 1 {X x M}P U x, P u a,b : ua,n ub,n, a,b R d,u x M O x M. For every F FCT, by 4..1 in [5], he damped gradien i defined by eq D FX[,T] x Q x, i U 1 i i fx x 1,,X x N, [,T]. eq eq i: i > Thu, he aociaed Mallavin gradien will defined a follow: D FX[,T] x i: i > I 1 {X xi M}P U xi U 1 i i fx x 1,,X x N, [,T]. For any conan K,K 1,σ,σ 1 wih K K 1,σ σ 1 and each [,T], le µ be he random meaure on [,T] given by µ dr exp[ K r σ l x r lx ]{ K 1 K dr + σ 1 σ dl x r }} : ϕ 1,r,K 1,K,σ 1,σ dr+ϕ 1,r,K 1,K,σ 1,σ dl x r Denoe by wo meaurable funcion on W T x M A 1+µ[,T] + B 1+µ[r,T] ϕ 1 r,,k 1,K,σ 1,σ dr 1+µ[r,T] ϕ r,,k 1,K,σ 1,σ dr. 3

4 Throughou he aricle, we aume ha he reflecing if M exi L-diffuion proce i non-exploive. Le P x be he diribuion of he L-diffuion proce X x aring from a fixed poin x up o ome fixed ime T >. Then P x i a probabiliy meaure on he Riemannian pah pace Wx T M. Define he following quadraic form by eq E K 1,K σ 1,σ F,F : D F A d+b dl dp x. Wx TM The following Logarihmic Sobolev inequaliy i he main reul of hi paper. T1.1 Theorem 1.1. Aume ha K Ric Z K 1 and σ I σ 1. Then he following Logarihmic Sobolev inequaliy hold eq E F F log E K 1,K F σ 1,σ F,F, F FCT. L By he above Theorem 1.1, we obain he following Corollary for wo pecial cae. C1. Corollary 1.. a Aume ha M i a Riemannian manifold wihou a boundary and K 1 Ric Z K, hen he Logarihmic Sobolev inequaliy hold eq E F F log CT,K F 1,K D F ddp x, F FC L Wx TM b M. In paricular, when K <, we have eq Spec L eq eq When K >, we have SG L 1 1+β 1+ K 1 K [ ] 1 e K T. K β + β β +β β e K T e K T, where β K 1 K K. b Le M be Ricci fla Riemannian manifold wih boundary, and we aume ha he econd fundamenal form aifie σ I σ 1, hen When σ, we ge ha for any F FCT E F log F F L W T x M 1+σ 1 lt x D F ddp x + σ 1 1+σ 1 l x Wx TM T T 4 D F dl dp x.

5 When σ <, we ge ha for any F FCT eq E F F log 1+ σ F 1 σ exp[ σ +εl x L Wx TM T ] + σ1 σ exp[ σ +εlt] x Wx TM for ome conan ε >. D F ddp x D F dl dp x Remark Wang [5] proved ha he damped Logarihmic Sobolev inequaliy. When M i a Riemannian manifold wihou boundary, and K Ric Z K 1,K +K 1, Fang-Wu [15] fir proved 1.1, laer, hi reul had been exended o he general cae of K and K 1 by Cheng-Thalimaier[5]. The re of hi paper i organized a follow: In Secion, we will prove Theorem 1.1 and Corollary 1.. The eimae of he pecral gap for he ochaic hea proce will be preened in Secion 3. Proof of Theorem 1.1 and Corollary 1..1 Proof of Theorem 1.1 Proof of Theorem 1.1. By Theorem 4.4 in [5], we know ha he following damped logarihmic Sobolev inequaliy hold eq.1.1 E F F log D F F ddp x. L Wx TM eq.. eq.3.3 Therefore i uffice o how ha D F d D F A d+b dl. By uing he aumpion of K Ric Z K 1 and σ I σ 1, we have Combining hi wih 1.14 Q x, I Ric Z K 1 K, I σ 1 σ, Q x,r{ Ric Z U x r d+iux r dlx r and [6, Theorem 3..1], i i eay o derive ha Q x,r exp[ K r σ l x r l x ]. 5 } I 1 {X x M}P U x.

6 By he definiion of he damped gradien, we ge eq.4.4 D FX[,T] x Q x, i U 1 i i fx x 1,,X x N D FX[,T] x i: i > i i: i > Q x,r{ Ric Z U x rd+iu x rdl x r} I 1 {X x M} P U x i U 1 i i fx x 1,,X x N D FX[,T] x Q,r{ x Ric Z Ur x D rfx[,t] x dr +IUx r D rfx[,t] r} x dlx. Then we have eq.5.5 D F X[,T] x D F X[,T] x + eq.6.6 exp[ K r σ l x r lx ]{ K 1 K dr+ σ 1 σ dl x r } D rf X x [,T] } D F X[,T] x + D r F µ dr. The Hölder inequaliy implie ha Thu, we obain eq D F d D F X x [,T] 1+µ[,T] r 1+µ [,T] D F + 1+µ[,T] D F d+ D F d+ 1+µ[,T] 1+µ[,T] D r F ϕ,r,k 1,K,σ 1,σ dl r d 1+µ[,T] D F d+ r 1+µ[,T] D r F ϕ,r,k 1,K,σ 1,σ dl r d 1+µ[,T] D F d+ 1+µ[r,T] ϕ r,,k 1,K,σ 1,σ dr D F dl D F A d+b dl. D r F µ dr 1+µ[,T] D r F µ drd D r F ϕ 1,r,K 1,K,σ 1,σ drd 1+µ[,T] D r F ϕ 1,r,K 1,K,σ 1,σ drd 1+µ[r,T] ϕ 1 r,,k 1,K,σ 1,σ dr D F d 6

7 eq.8.8 Up o now, we complee he proof.. Proof of Corollary 1. To prove Corollary 1., we need ome preparaion. Le β K 1 K K and define [ ] Λ,T 1+β 1 exp[ K T ] + β +β [ ] 1 exp K and + β [exp K +T exp K T ] CT,K 1,K : up Λ,T. [,T] Similar o he proof of Propoiion 3.3 in Fang-Wu[15] for he cae of K 1 +K, in he following we will dicu monooniciy of he funcion Λ, T. p.1 Propoiion.1. 1 If K <, hen Λ,T i ricly increaing over [,T]. If K, hen he maximum i aained a a poin in,t. Proof. According o he definiion.8 of Λ,T, we ge Λ,T 1+β 1 e K T and ΛT,T 1+ β +β [ 1 exp K T ]+ β [ e K T 1 ] [ ] 1+β 1 e K T Λ,T. eq.9.9 In paricular, he econd in he above implie ha ΛT,T Λ,T. Nex, we ake he derivaive of Λ,T wih repec o, Λ,T βk e KT + β +β K e K β K [ e K +T +e ] K T. Then we have and eq.1.1 Λ,T βk e K T + β +β K β K e K T βk 1+β1 e K T ; Λ T,T βk + β +β K e KT β K [ e K T +1 ] βk 1 1+βe KT + β [ e K T +1 ] βk [1 e K T ] β K [1 e K T ]. 7

8 Noing ha eq { Λ T,T > if K <, Λ T,T < if K >. Now we look for [,T] uch ha Λ,T. We have Λ,T eq.1.1 e KT +1+βe K β [ e K +T +e ] K T e KT e K +1+β β [ e K T +e KT e ] K 1+ β e KT e K 1+β β e K T. Therefore here exi a mo one uch ha Λ,T. For he cae where K <, if here exi,t uch ha Λ,T <. Then by.9 and.11, he equaion Λ,T ha a lea wo oluion, i i impoible. Therefore for K <, Λ,T. For K >, we uppoe uch ha Λ,T, hen by.1 e K Thu he proof i compleed. 1+ β 1 e K T e KT. +β By he above Propoiion.1, i i eay o obain he following Propoiion.. p. Propoiion.. i If K >, eq up Λ,T 1+β [,T] β + β 1+ β +β β +β β e K T 1+ β +β 1 e K T T e K. 1 e K T e K T eq ii If K <, up [,T]Λ,T K 1 K [ ] 1 e K T. K 8

9 Proof of Corollary 1.. a Since M i a Riemannian manifold wihou boundary, hen he local ime l, hu we have Then µ dr : K 1 K exp[ K r ]dr. eq µ[,t] K 1 K [1 exp[ K T ]]. K Which implie ha eq ϕ 1 r,,k 1,K,σ 1,σ K 1 K exp[ K r] ϕ r,,k 1,K,σ 1,σ. Then by.15 and he fir equaliy of.16, eq µ[r,T] ϕ 1 r,,k 1,K,σ 1,σ dr K 1 K 1+ K 1 K [1 exp[ K T r]] exp[ K r]dr K K1 K + K 1 K [ 1 exp K K K ] K 1 K exp K K +TexpK 1. Thu we ge eq B A 1+µ[r,T] ϕ r,,k 1,K,σ 1,σ dr 1+µ[,T] + 1+µ[r,T] ϕ 1 r,,k 1,K,σ 1,σ dr 1+ K 1 K [1 exp[ K T ]] K K1 K + + K 1 K [ 1 exp K K K ] + K 1 K [exp K K +T exp K T ]. From which we have eq D F A d+b dl Λ,T D F d. 9

10 Then 1.9, 1.1 and 1.11 come from Theorem 1.1 and Propoiion.. b By he aumpion of M, we know ha eq.. µ dr σ 1 σ exp[ σ lr x lx ]dlx r. Thu, eq.1.1 µ[,t] σ 1 σ exp[σ l x ] exp[ σ lr x ]dlx r. In addiion,. implie ha eq.. ϕ 1 r,,k 1,K,σ 1,σ ϕ r,,k 1,K,σ 1,σ σ 1 σ exp[ σ l x lx r ]. Then, by he definiion of A and B, eq.3.3 A 1+ σ 1 σ exp[σ l x ] exp[ σ lr x ]dlx r B σ 1 σ 1+ σ 1 σ exp[σ l xr ] exp[ σ lu x ]dlx u exp[ σ l x lx r ]dr. r eq.4.4 Since l x i a increaing proce, we have A { 1+σ 1 l x T, if σ, 1+ σ 1 σ exp[ σ +εl x T ], if σ < and eq.5.5 B { σ 1 1+σ 1 l x T T, if σ, σ 1 σ exp[ σ +εl x T ], if σ < By Theorem 1.1, when σ, we ge E F F log F L eq σ 1 lt x W T x M + σ 1 1+σ 1 l x Wx TM T T D F ddp x D F dl dp x, 1

11 and when σ <, we ge E F F log F L eq σ 1 σ exp[ σ +εl x Wx TM T ] D F ddp x + σ1 σ exp[ σ +εlt] x D F dl dp x. W T x M eq.8.8 Corollary.3. Le M : { x x 1,,x d R d : a 1 x 1 + +a d x d c } for ome conan c R, hen M i a Riemannian manifold wihou boundary and Ric wih I, hu we have E F log F F L 3 Sochaic hea equaion D F ddp x. Wx TM In hi ecion, we will conider he pecral gap for he ochaic hea equaion on a Riemannian manifold wih boundary. Before moving on, le inroduce ome noaion. The ochaic hea equaion on Riemannian manifold had been udied deailed by []ee alo [17]. Here hey inroduced ome noaion. In paricular, he claical cylinder funcion depending on finie ime i no in he domain of generaor aociaed o he ochaic hea equaion. Thu, we need o inroduce a cla of new cylinder funcion FCb 1 on WT x M, i.e. for every F FC1 b, here exi ome m 1, m N, f Cb 1Rm,g i C,1 b [,1] M, i 1,...,m, uch ha 1 eq Fγ f g 1,γ d, 1 g,γ d,..., 1 g m,γ d, γ Wx T M, where C,1 b [,1] M denoe he funcion which are coninuou w.r.. he fir variable and differeniable w.r.. he econd variable wih coninuou derivaive. For any F FC 1 b wih. form and h L [,1];R d, according o Wang[5], he damped Malliavin gradien of F i given by DFγ : m j1 ˆ j fγ Q,u Uu 1 γ g j u,γ u du, γ Wx T M. 11

12 Le F be he damped L -gradien of F, and ince Then, we have Thu, DF,h F,h d d h,df H D h F h,df L F, h udu d F,h u dud Fudu,h d. F m j1 The L -gradien of F i defined by Fudu DF. u F,h u ˆ j fγq,t U 1 γ g j,γ. ddu F m j1 ˆ j fγu 1 γ g j,γ. By Lemma 4.3. in [5], we have eq3. 3. F F + F + F + DF,dB Fudu,dB Q,u Fudu,dB eq By he andard he procedure, we have E F log F F L Wx TM Q,u Fudu ddp x. 1

13 eq By.3 and Hölder inequaliy, we ge Le Q,u Fudu ϕ e Ku σlx u l x Fudu e Ku σlx u lx du e Ku σlx u lx du e Ku σlx u lx du. Then by changing he order of inegraion we obain W T x M where W T x M Q,u Fudu A F ddp x, A ddp x W T x M ϕue K u σlx lx u du. Thu, we ge he following Logarihmic Sobolev inequaliy. e Ku σlx u lx F udu e Ku σlx u lx F udu. ϕ e Ku σlx u lx F ududdp x T3.1 Theorem 3.1. Aume ha Ric Z K and I σ. Then he following Logarihmic Sobolev inequaliy hold eq E F log F F L A F ddp x, F FCb. 1 Wx T M c3. Corollary 3.. a Aume ha M i a Riemannian manifold wih a convex boundary and K Ric Z, hen he Logarihmic Sobolev inequaliy hold eq E F log F F L CT,K F ddp x, F FC Wx TM b M, where [ ] 1 e KT K CT,K e e KT e KT 1, if K, KT 1 [ 1 K 1 e KT ], if K <. 13

14 Proof. Since he boundary i convex, hu I σ. Thu and eq ϕ Then we ge e Ku σlx u lx du e Ku du 1 K [ 1 e KT ] A ϕue K u σlx l u x du 1 [ ] 1 e KT u e K u du K 1 ] [ e K +e KT+ e KT. K A, AT 1 1 e K K In he following, imilar o he argumen of Propoiion.1. Taking he derivaive of A give A 1 ] [e K e KT+ e KT. K Thu, Noing ha A 1 K [ ] 1 e K, A T 1 [ e K 1 ]. K eq { A T > if K <, A T < if K >. Now we look for [,T] uch ha A. We have c A e K e KT+ e KT e KT e KT+K e K e KT 1. Therefore here exi a mo one uch ha A. For he cae where K <, if here exi,t uch ha A <. Then by 3.7 and 3.8, he equaion A ha a lea wo oluion, i i impoible. Therefore for K <, A. For K >, we uppoe uch ha Λ,T, hen by 3.9 The proof i compleed. e K e KT 1. 14

15 Corollary 3.3. Le M be a Ricci-fla Riemannian manifold wih a convex boundary, hen we have E F F log T F ddp F x. L Reference W T x M [1] S. Aida and K. D. Elworhy, Differenial calculu on pah and loop pace. I. Logarihmic Sobolev inequaliie on pah pace, C. R. Acad. Sci. Pari Série I, , [] D. Bakry, M. Ledoux, F.-Y. Wang, Perurbaion of funcional inequaliie uing growh condiion, J. Mah. Pure Appl. 877, [3] B. Capiaine, E. P. Hu and M. Ledoux, Maringale repreenaion and a imple proof of logarihmic Sobolev inequaliie on pah pace, Elecron. Comm. Probab. 1997, [4] X. Chen, B. Wu, Funcional inequaliy on pah pace over a non-compac Riemannian manifold, J. Func. Anal. 6614, [5] L. J. Cheng, A. Thalmaier, Specral gap on Riemannian pah pace over aic and evolving manifold, J. Func. Anal V984 [6] A. B. Cruzeiro and P. Malliavin, Renormalized differenial geomery and pah pace: rucural equaion, curvaure, J. Func. Anal. 139: , [7] B. K. Driver, A Cameron-Marin ype quai-invariance heorem for Brownian moion on a compac Riemannian manifold, J. Func. Anal , [8] B. K. Driver and M. Röckner, Conrucion of diffuion on pah and loop pace of compac Riemannian manifold, C. R. Acad. Sci. Pari Série I , [9] K. D. Elworhy, X.- M. Li and Y. Lejan, On The geomery of diffuion operaor and ochaic flow, Lecure Noe in Mahemaic, , Springer-Verlag. [1] K. D. Elworhy and Z.-M. Ma, Vecor field on mapping pace and relaed Dirichle form and diffuion, Oaka. J. Mah , [11] O. Enchev and D. W. Sroock, Toward a Riemannian geomery on he pah pace over a Riemannian manifold, J. Func. Anal. 134 : 1995, [1] S.- Z. Fang, Un inéqualié du ype Poincaréur un epace de chemin, C. R. Acad. Sci. Pari Série I , [13] S.- Z. Fang and P. Malliavin, Sochaic analyi on he pah pace of a Riemannian manifold: I. Markovian ochaic calculu, J. Func. Anal. 118 : , [14] S.- Z. Fang, F.-Y. Wang and B. Wu, Tranporaion-co inequaliy on pah pace wih uniform diance, Sochaic. Proce. Appl. 118: 1 8, [15] S. Z. Fang, B. Wu, Remark on pecral gap on he Riemannian pah pace, Elecron. Commun. Probab. 17, no. 19, 1V13. 15

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