ESTIMATES FOR THE DERIVATIVE OF DIFFUSION SEMIGROUPS

Transcription

1 Elec. Comm. in Probab. 3 (998) ELECTRONIC COMMUNICATIONS in PROBABILITY ESTIMATES FOR THE DERIVATIVE OF DIFFUSION SEMIGROUPS L.A. RINCON Deparmen of Mahemaic Univeriy of Wale Swanea Singleon Par Swanea SA2 8PP, UK L.Rincon@wanea.ac.u ubmied April 5, 998; revied Augu 8, 998 AMS 99 Subjec claificaion: 47D7, 6J55, 6H. Keywor and phrae: Diffuion Semigroup, Diffuion Procee, Sochaic Differenial Equaion. Abrac Le {P } be he raniion emigroup of a diffuion proce. I i nown ha P en coninuou funcion ino differeniable funcion o we can wrie DP f. Bu wha happen wih hi derivaive when and P f = f i only coninuou?. We give eimae for heupremumnormofhefréche derivaive of he emigroup aociaed wih he operaor A + V and A + Z where A i he generaor of a diffuion proce, V i a poenial and Z i a vecor field. Inroducion Conider he following ochaic differenial equaion on R n dx = X(X ) db + A(X ) d, X = x R n, for, where he fir inegral i an Iô ochaic inegral and he econd i a Riemann inegral. Here {B } i Brownian moion on R m and he equaliy hol almo everywhere. The coefficien of hi equaion are he mapping X: R n L(R m ; R n ) and he vecor field A: R n R n. Aume andard regulariy condiion on hee coefficien o ha here exi a rong oluion {X } o our equaion. We wrie X x for X when we wan o mae clear i dependence on he iniial value x. I i nown ha under furher aumpion on he coefficien of our equaion, he mapping x X x i differeniable ( ee for inance [] ). Suppored by a gran from CONACYT (México). 65

2 66 Elecronic Communicaion in Probabiliy Aume coefficien X and A are mooh enough and conider he aociaed derivaive equaion dv = DX(X )(V ) db + DA(X )(V ) d, V = v R n, whoe oluion {V } i he derivaive of he mapping x X x a x in he direcion v. We will aume ha here exi a mooh map Y: R n L(R n ; R m ) uch ha Y(x) iherigh invere of X(x). Tha i, X(x)Y(x) =I R n for all x in R n. We hall alo aume ha he proce {Y(X )(V )} belong o L 2 ([,]) for each >, ha i, Y(X )(V ) 2 < and hu we can wrie Y(X )(V ) db. Wih all above aumpion, we have from [6], he following reul (ee Appendix for a proof) Theorem For every > here exi poiive conan and a uch ha E Y(X )(V ) db e a. () We are inereed in mall value for. Thu, ince e a =O( )a, we have ha here exi a poiive conan N uch ha e a N for mall. Hence for ufficienly mall, we have he following eimae E Y(X )(V ) db c, (2) where c i a poiive conan. Le now BC r (R n ) be he Banach pace of bounded meaurable funcion on R n which are r-ime coninuouly differeniable wih bounded derivaive. The norm of hi pace i given by he upremum norm of he funcion plu he upremum norm of each of i r derivaive. In paricular B(R n ) i he Banach pace of bounded meaurable funcion on R n wih upremum norm =up x R n f(x). Suppoe our diffuion proce {X } ha raniion probabiliie P (, x, Γ). Then hi induce a emigroup of operaor {P } a follow. For every we define on B(R n ) he bounded linear operaor (P f)(x) = f(y) P (, x, dy) =E(f(X x )). (3) R n The emigroup {P } i a rongly coninuou emigroup on BC (R n ). Denoe by A i infinieimal generaor. I i nown ha {P } i a rong Feller emigroup, ha i, P en coninuou funcion ino differeniable funcion. In fac, under above aumpion, a formula for he derivaive of P f i nown ( ee [4] or [5] ). Theorem 2 If f BC 2 (R n ) hen he derivaive of P f : R n R i given by D(P f)(x)(v) = E{f(X ) Y(X )(V ) db }. (4)

3 Diffuion Semigroup 67 Higher derivaive in a more general eing are given in [4]. See alo [2] for a general formula of hi derivaive in he conex of a ochaic conrol yem. Oberve he mapping f E{f(X ) Y(X )(V ) db } define a bounded linear funcional on BC 2 (R n ). Hence here exi a unique exenion on BC (R n ). Since he expreion of hi linear funcional doe no depend on he derivaive of f, i ha he ame expreion for any f in BC (R n ). From la heorem we obain DP f(x)(v) E Y(X )(V ) db e a. And hence for mall DP f c. (5) Oberve ha, a expeced, our eimae goe o infiniy a approache ince P f = f i no necearily differeniable. Alo he rae a which i goe o infiniy i no faer han doe. 2 Poenial Le V : R n R be a bounded meaurable funcion. We hall perurb he generaor A by adding o i he funcion V. We define he linear operaor A V = A + V, wih he ame domain a for A. A emigroup {P V } having A V a generaor i given by he Feynman-Kac formula P V f = E{f(X )e V (Xu) du }. We will find a imilar eimae a (5) for DP V f. We fir derive a recurive formula ha will help u calculae he derivaive of P V f. Wehave Hence P V f = P f + [ P P V f ] = = P f + = P f + = (P P V f) [ A(P P V f)+p ((A + V )P V f)]. P V f = P f + P (VP V f).

4 68 Elecronic Communicaion in Probabiliy Now we ue our formula for differeniaion (4) o calculae he derivaive of hi emigroup. We have DP V f(x)(v) = DP f(x)(v) + DP (VP V f)(x)(v) = E{f(X ) Y(X )(V ) db } + E{V (X )P V f(x ) Y(X u )(V u ) db u }. Then, by he Feynman-Kac formula and he Marov propery we have DP V f(x)(v) = E{f(X ) + from which we obain And hence for mall Y(X )(V ) db } E{V (X )E{f(X )e DP V f E Y(X )(V ) db + V DP V f c V (Xu)du } e V E + V e V Y(X u )(V u ) db u }, Y(X u )(V u ) db u. = c +2c V e V. Oberve again ha our eimae goe o infiniy a. 3 Bounded Smooh Drif c Le Z : R n R n be a bounded mooh vecor field. We hall conider anoher perurbaion o he generaor A. Thi ime we define he linear operaor A Z = A + Z. The exience of a emigroup {P Z } having A Z a infinieimal generaor i guaraneed by he regulariy of Z. Indeed, if we wrie Z(x) =(Z (x),...,z n (x)), hen he operaor A Z can be wrien a A Z = 2 n (X(x)X(x) ) ij 2 x i x + n j i,j= i= (A i (x)+z i (x)) x i, and hi operaor i he infinieimal generaor aociaed wih he equaion dx = X(X )db +[A(X )+Z(X )]d, X = x R n.

5 Diffuion Semigroup 69 Than o he moohne of Z, hi equaion yiel a diffuion proce (X x,z ) T and hence he emigroup P Z f(x) =E(f(Xx,Z )). Then previou eimae applie alo o DP Zf. Bu we can do beer becaue we can find he explici dependence of he eimae upon Z a follow. A before, we fir find a recurive formula for hi emigroup. Hence P Z f = P f + [ P P Z f ] = = P f + = P f + = (P P Z f) [ A(P P Z f)+p ((A + Z )P Z f)]. P Z f = P f + We can now calculae i derivaive a follow DP Z f(x)(v) = DP f(x)(v) + P (Z P Z f). (6) DP (Z P Z f)(x)(v) = E{f(X ) Y(X )(V ) db } + E{Z(X ) P Z f(x ) Y(X u )(V u ) db u }. We now find an eimae for he upremum norm of hi derivaive. Taing modulu we obain DP Z f(x)(v) E Y(X )(V ) db + E DP Z f(x )(Z(X )) Y(X u )(V u ) db u. Hence for mall DP Z f c DP Z f + c Z f. We now olve hi inequaliy. If we ierae once we obain DP Z f c + c 2 Z + c 2 Z 2 By Fubini heorem, he double inegral become DP Z u f u ( ) DP Z u f ( )( u) du. ( )( u) du,

6 7 Elecronic Communicaion in Probabiliy andhenweoberveha u u =2an ( )( u) u = π. The cae u = olve alo he fir inegral. Hence our inequaliy reduce o DP Z f c + c 2 π Z + c 2 π Z 2 DP Z u f du. We now apply Gronwall inequaliy. Afer ome implificaion ( exending he inegral up o infiniy ) we finally obain he eimae DP Z f c +2c 2 π Z e (c2 π Z ) (7) A expeced, our eimae goe o infiniy a incep Z f = f i no necearily differeniable. 4 Bounded Uniformly Coninuou Drif We now find a imilar eimae when Z : R n R n i only bounded and uniformly coninuou. We loo again a he operaor A Z = A + Z. The problem here i ha in hi cae we do no have he emigroup {P Zf} ince he ochaic equaion wih he added nonmooh drif Z migh no have a rong oluion. So we canno even al abou i derivaive. To olve hi problem we proceed by approximaion. 4. Exience of Semigroup Since Z BC (R n ; R n ) i uniformly coninuou, and BC (R n ; R n )ideneinbc (R n ; R n ), here exi a equence {Z i } i= in BC (R n ; R n ) uch ha Z i converge o Z uniformly. Thu, for every i N, wehaveheemigroup{p } ince our ochaic equaion wih he added mooh drif Z i ha a rong oluion. For every andf BC (R n ) fixed, he equence of funcion {P f} i= i a Cauchy equence in he Banach pace BC (R n ). We will prove hi fac laer. Le u denoe i limi by P Z f. All properie required for P Z f are inheried from hoe of he emigroup P f. Indeed by imply wriing P Z f = lim i P f and uing an inerchange of limi we can prove. f P Z f i a bounded linear operaor. 2. P Z f i a conracion emigroup of operaor. 3. {P Z } ha generaor A Z = A + Z. Now, uppoe for a momen ha he equence of derivaive {DP f} i= converge uniformly. Then we would have D(lim i P f) = lim i DP f. Thi prove ha P Z f i differeniable. Then, for any ɛ>hereexin N uch ha if i N, DP Z f DP f +ɛ and herefore our eimae alo applie o DP Z f.

7 Diffuion Semigroup Uniform Convergence of Derivaive We now prove ha he equence {DP formula (6) o obain P f = f} i= converge uniformly. We ue again our recurive P (Z i P f Z j P Zj f). We now ue our formula for differeniaion (4). Afer differeniaing and aing modulu we obain D(P f)(x)(v) E{ DP f(x )(Z i (X )) DP Zj f(x )(Z j (X )) Y(X u )(V u ) db u }. Thu D(P f) Z i Z j + Z j And hence for mall D(P f) Z i Z j + Z j Now wrie eimae (7) a e a( ) DP e a( ) DP DP f DP DP f A + Be C, f f DP Zj f. f DP Zj f. for ome poiive conan A, B and C depending on and Z i. Thu, ubiuing hi in our la eimae give A D(P f) Z i Z j + Z i Z j Be C + Z j DP Zj f DP f. The fir wo inegral are bounded if we allow o move wihin a finie inerval (,T]. Thu here exi poiive conan M and M 2 uch ha D(P f) M Z i Z j + M 2 DP f DP Zj f.

8 72 Elecronic Communicaion in Probabiliy Now ierae hi o obain D(P f) M Z i Z j A before he double inegral reduce o πm2 2 + M 2 M Z i Z j + M 2 2 D(Pu f P u Zj f) du. Collecing conan ino new conan M and N we arrive a D(P f) M Z i Z j + N Now we apply again Gronwall inequaliy o obain D(Pu f Pu Zj f) ( du. )( u) D(Pu f P u Zj f) du. D(P f) M Z i Z j + NM Z i Z j e N( u) du. The righ hand ide goe o zero a Z i Z j. Thi prove he equence {DP } i= i uniformly convergen. 4.3 Uniform Convergence of Semigroup We finally prove ha he equence of funcion {P recurive formula (6) we obain f} i= i a Cauchy equence. From our P f Z i P + f Z j P Zj f Z i Z j DP f Z j DP f DP Zj f Z i Z j e a + Z j DP f DP Zj f. The fir inegral i bounded o he fir par goe o zero a Z i Z j approache. The econd par alo goe o zero ince we ju proved he equence of derivaive i a Cauchy equence. Hence {P f} i= i uniformly convergen. Thu, wih hi approximaing procedure we found a emigroup for he operaor A + Z for Z bounded and uniformly coninuou and we proved i i differeniable and ha our eimae alo applie o i derivaive.

9 Diffuion Semigroup 73 Oberve ha he boundedne and uniform coninuiy of Z are required in order o enure he exience of a equence of mooh vecor fiel Z i uniformly convergen o Z. Alernaive aumpion on Z ha guaranee he exience of uch approximaing equence may be ued. Appendix We here give a proof of inequaliy () which i aen from [6]. By uing he Cauchy-Schwarz inequaliy and hen he iomeric propery, we have E Y(x )(v ) db ( E = (E Y 2 ( Y(x )(v ) db 2 ) /2 Y(x )(v ) 2 ) /2 E v 2 ) /2. (8) We will eimae he righ-hand ide of he la inequaliy. Iô formula applied o he funcion f( ) = 2 : R n R and he emimaringale (v ) yiel n v 2 = v 2 +2 v u dv u + d v i u. Therefore i= E v 2 = v 2 + E v = v 2 + E [DX(x u )(v u )] [DX(x u )(v u )] du = v 2 + E v 2 + DX 2 We now ue Gronwall inequaliy o obain Inegraing from o yiel and hu, ubiuing in (8) we obain E Thi prove inequaliy (). DX(x u )(v u ) 2 du E v u 2 du. E v 2 v 2 e DX. E v 2 v 2 DX (e DX ), Y(x )(v ) db Y 2 v DX /2 e DX.

10 74 Elecronic Communicaion in Probabiliy Acnowledgmen I would lie o han Prof. K. D. Elworhy a Warwic from whom I have enormouly benefied. Alo han o Dr. H. Zhao and Prof. A. Truman a Swanea for very helpful uggeion. CONACYT ha financially uppored me for my pograduae udie a Warwic and Swanea. Reference [] Da Prao, G. and Zabczy, J. (992) Sochaic Equaion in Infinie Dimenion. Encyclopedia of Mahemaic and i Applicaion 44. Cambridge Univeriy Pre. [2] Da Prao, G. and Zabczy, J. (997) Differeniabiliy of he Feynman-Kac Semigroup and a Conrol Applicaion. Rend. Ma. Acc. Lincei. 9, v. 8, [3] Elworhy, K. D. (982) Sochaic Differenial Equaion on Manifol. Cambridge Univeriy Pre. [4] Elworhy, K. D. and Li, X. M. (994) Formulae for he Derivaive of Hea Semigroup. Journal of Funcional Analyi 25, [5] Elworhy, K. D. and Li, X. M. (993) Differeniaion of Hea Semigroup and applicaion. Warwic Preprin 77/993. [6] Li, X.-M. (992) Sochaic Flow on Noncompac Manifol. Ph.D. Thei. Warwic Univeriy. [7] Pazy, A. (983) Semigroup of Linear Operaor and Applicaion o Parial Differenial Equaion. Springer-Verlag. New Yor. [8] Rincón, L. A. (994) Some Formulae and Eimae for he Derivaive of Diffuion Semigroup. Warwic MSc. Dieraion.