Transformations of measure on infinite-dimensional vector spaces

Transcription

1 Transformaions of measure on infinie-dimensional vecor spaces Denis Bell Deparmen of Mahemaics, Universiy of Norh Florida 4567 S. Johns Bluff Road Souh,Jacksonville, FL 32224, U. S. A. This paper has appeared in he volume Seminar on Sochasic Processes, Vancouver, 199. Birkhauser, Boson,

2 2 Denis Bell 1. Inroducion Le denoe a Banach space equipped wih a finie Borel measure ν, T : a measurable ransformaion of, and ν T he image measure of ν under T, defined by ν T (B) =ν(t 1 (B)) for Borel ses B. Aransformaion heorem for ν is a resul ha gives condiions on T under which he measures ν T and ν are muually absoluely coninuous, and a formula for he corresponding Radon-Nikodym derivaives (RND) when hese condiions hold.if is finie-dimensional hen he problem of rnsformaion of measure falls wihin he scope of he change of variables (Jacobi) heorem. In he infinie-dimensional case he problem is more suble. In paricular, here exiss no ranslaion-invarian reference measure. In his seing he sudy of ransformaion of measure has largely been resriced o cases where he measure is Gaussian. In his aricle we discuss a scheme inroduced by he auhor in [B.2, Secion 2] for obaining ransformaion heorems for arbirary Borel meaures defined on (finie or) infinie-dimensional vecor spaces. Alhough formal, i is hoped ha his procedure can be made rigorous in specific cases where addiional srucure is assumed. In he general seing discussed here, i yields a formula for he RND dν T /dν ha we believe o be new. 2. Transformaion heorems for Gaussian measure These come in wo varieies, classical and absrac. The heory of ransformaion of he classical Wiener measure was developed by Cameron and Marin, and Girsanov. Girsanov s heorem is as follows. Theorem 1 (Girsanov s heorem) Le w denoe a sandard real-valued Wiener process and ν he law of w (i.e. he Wiener measure). Le h be a bounded measurable process adaped o he filraion of w and consider he process y defined by y = w h s ds, [, 1]. Then y is a sandard Wiener process wih respec o he measure dµ(w) G(w)dν(w), where G(w) exp h s dw s 1 1 h 2 2 sds. (1) A series of (increasingly more general) resuls have been proved concerning he ransformaion of absrac Gaussian measure. The quinessenial paper in his area is due o Ramer [R]. Le (i, H, ) denoe an absrac Wiener spaces as defined by Gross [G]. Le ν denoe he corresponding Gaussian measure on, and denoe by <.,.>and., he inner produc and norm on H, respecively.

3 Transformaions of measure 3 Theorem 2 (Ramer) Suppose U is an open subse of and T : U is a map of he form I + K, where I is he ideniy map on. Suppose (i) T is a homeomorphism from U o T (U). (ii)k:u H is an H C 1 map such ha he map x U DK(x) is coninuous ino he space of Hilber-Scmid operaors on H. (iii) I H + DK(x) GL(H), for every x. Then he meaure ν T 1 is absoluely coninuous wih respec o ν and dν T 1 dν (x) = δ(dt(x)) exp { <K(x),x> race H DK(x) K(x) 2 /2 (2) where δ denoes he Carleman-Fredholm deerminan defined on L(H). (The difference of he wo expressions conained in he quoe marks in (2) is defined as he limi of a convergen sequence in L 2 of differences. Boh of he erms may fail o exis separaely). In [B.1], he auhor proved he following non-gaussian version of he Cameron- Marin heorem (cf. [K.2]). Theorem 3 (Bell) Le ν denoe a finie Borel measure on saisfying he following condiion wih respec o a vecor r D r φdν = φx r dν. for es funcions φ defined on. Suppose he funcion R X r (x + r) is coninuous, a. e. x and he following random variables are locally inegrable sup { X r (x + r) 4, 1, { sup exp 4 s X r (x + ur)du. Define T (x) =x r. Then he measures ν T and ν are equivalen and dν T dν (x) = exp 1 X r (x + ur)du. (4) 3. Transformaions of measure via a homoopic consrucion Le ν be a finie Borel measure on a Banach space and le U denoe a disinguished subclass of he class of ransformaions of. Definiion A linear operaor L : U L 2 (ν) isaninegraion by pars operaor (IPO) for ν if he following holds for all C 1 funcions φ : R and all h U for which boh sides exis Dφ(x)h(x)dν = φ(x)lh(x)dν.

4 4 Denis Bell Remark. The Malliavin calculus provides a mehod for obaining IPOs for measues induced by sochasic differenial equaions (cf. [B.2, Chapers 2-4 and Secion 7.3). Suppose ha L is an IPO for he measure ν wih domain U. The following resul is easily verified (cf. [B.2, Secion 5.3]) Lemma Le h U L 2 (ν),ψ : R L 2 (ν) C 1,ψh U. Then L(ψh)(x) =ψ(x)lh(x) Dψ(x)h(x) a.s.ν. Remark. The above lemma can be shown o hold for a larger se of funcions h and ψ by a closure argumen. Le T : denoe a map of he form T = I + K where K U. Define T = I + k, [, 1]. Suppose T is inverible for all [.1]. Noe ha he ransformaions T are absoluely coninuous wih respec o ν if and only if here exiss a family {X : [, 1] of RNDs such ha X = 1 and for all es funcions φ on, we have φ(x)dν = φ T 1 (x)x (x)dν. (5) Thus he RHS of (5) is independen of. Differeniaion wr under he inegral sign gives { Dφ(T 1 (x))(d/dt 1 (x))x (x)+φ T 1 (x)d/dx (x) dν =. (6) We simplify he firs erm in he inegrand in (6) by means of he relaion Dφ(T 1 (x))d/dt 1 (x) = D(φ T 1 )(x)k T 1 (x) o ge { φ T 1 d/dx (x) D(φ T 1 )(x)k T 1 (x)x (x) dν =. (7) Asume ha (K T )X U. Then he defining propery of L allows us o wrie (7) in he form φ T 1 (x) { d/dx (x) L[(K T 1 )X ](x) dν.

5 Transformaions of measure 5 This will hold for all es funcions φ if and only if X saisfies he differenial eq uaion d/dx (x) =L[(K T 1 )X ](x). (8) Now assume he funcions K T 1 and h saisfy he condions on h and ψ respecively in he above lemma. Then applying he lemma o he RHS of (8) yields d/dx (x) =X (x)l[(k T 1 ](x) DX (x)k T 1 (x). (9) Wriing X(, x) =X (x), X 1 = dx /d, X 2 = DX, and subsiuing x = T (y) in (9) gives X 1 (, T (y)) = X(, T (y))l[k T 1 ](T (y) X 2 (, T (y))k(y). However, since K = DT /d, his is equivalen o dx /dx(, T (y)) = X(, T (y))l[k T 1 ](T (y). (1) Solving equaion (1) wih he iniial condiion X(,x) = 1 gives { X(, T (y)) = exp L[K Ts 1 ](T s (y))ds. We hus arrive a he following expression for X { X(, x) = exp L[K Ts 1 ](T s T 1 (x))ds. (11) In paricular dν T dν (x) =X(, x) = exp L[K Ts 1 ](T s T 1 (x))ds. (12) Suppose one is given a measure ν on, an IPO L for ν wih domain U, and a map T of of he form I + K wih K U such ha he maps T = I + K are inverible for all [.1]. Then one could obain a ransformaion heorem for ν by defining X(, x) by he formula (11) and esablishing (5) by reversing he seps in he above argumen. This will imply he equivalence of he measures ν T and ν, wih X as he corresponding RNDs. This scheme was implemened in [B.2] in he case K is consan o derive Theorem 3 above. We now give a condiion on K which ensures he inveribily of he maps T Definiion The map K is said o be a (srong) conracion on if here exiss a consan c [, 1) such ha K(x) K(y) c x y, x, y.

6 6 Denis Bell Proposiion If K is a conracion on hen he ransformaion T s = I + sk is inverible, for all s [, 1]. Proof. I suffices o prove he resul for T = I + K. The conracion propery rivially implies ha T is injecive. To show T is surjecive, suppose y and define K y (x) y K(x). Then K y is a conracion on. By he conracion mapping heorem, K y has a fixed pf x. Then x saisfies T (x )=y. 4. Transformaion formulae In his secion, we use (12) o derive he formulae for he densiies ha occur in Theorems1, 2, and 3. (A) Le ν denoe he sandard Wiener measure on he space of real-valued pahs wih iniial poin, defined on he ime inerval [.1]. Then he Iô inegral operaor L, Lk 1 k sdw s (13) is an IPO for ν. The domain U of L consiss of adaped pahs k lying in he Cameron-Marin space, i.e. such ha k = and 1 k s 2 ds <. This resul is due o Gaveau & Trauber[G-T]. Le h = h(w) denoe a bounded adaped pah as in he saemen of Girsanov s heorem and define. K(w) h u du. Subsiuing he operaor L definined in (13) ino (12), we have dν T dν T (w) = exp = exp 1 1 ( h u d w u s h u dw u s = exp h u dw u 1 2 =1/G(w) 1 1 u ) h 2 udu h 2 udu h v dv ) ds where w is as in (1). Thus we obain he formula for he densiy in Girsanov s heorem.

7 Transformaions of measure 7 (B) Le (i, H, ) denoe an absrac Wiener space wih Gaussian measure ν on. The Gaussian divergence operaor 1 LK(x) < K(x),x> race H DK(x) where <.,.>is he inner produc on H, is an IPO for ν (cf., e.g. K.2]). The domain of L can be chosen o be he se of C 1 funcions from ino. where is idenified wih is image in under he inclusions i i H = H. This domain can hen be exended o he se U conising of he class of maps K : H in Theorem 2, using he argumen in Ramer s paper [R]. For K U, one hen has LK(x) = <K(x),x> race H DK(x) (14) where he quoe marks have he same meaning as in [R]. In order o derive he densiy formula (2) from (12) and (14), i will be necessary o perform some manipulaions on he race erm in (14). In he presen generaliy hese manipulaions are necessarily of a formal naure, since he race may fail o exis as a separae eniy (his problem could be circumvened by performing he manipulaions on he sequence of approximaions used o define he random variable on he RHS in (14), in he spiri of Ramer [R], hen passing o he limi). In order o avoid his problem, we make he sronger assumpion ha K is a C 1 m,ap ino. In his case, boh erms on he RHS in (14) exis and we have 1 L[K T 1 s ](T s (x))ds = exp <K(x),x>+ K(x) 2 /2 race H D[K Ts 1 ](T s (x))ds = DeDT (x) 1 exp { <K(x),x>+ K(x) 2 /2 (15) where (15) follows freom he ideniy 1 exp {race H D[K Ts 1 ](T s (x))ds = DeDT (x). We obain from (15) dν T 1 dν (x) = DeDT (x) exp { <K(x),x>+ K(x) 2 /2. (16) 1 In he case where he absrac Wiener space is he classical Wiener space, he operaor L can be used o define an exension of he Iô inegral wih anicipaing inegrands. This exension is known as he Skorohod inegral.

8 8 Denis Bell This is he formula for he RND in a ransformaion formula due o Kuo [K.1]. To derive he analogous formula (2) in Ramer s heorem, i is necessary o inroduce he (formal) quaniy race H DK(x) ino he exponenial in (16). The corresponding adjusmen ouside he exponenial convers he sandard deerminan in (16) ino he Carlkeman-Fredholm deerminan and gives rise o he formula (2). (C) Suppose ν is a finie Borel measure on such ha (3) holds for some r. Define U {λr : λ R and L on U by L(λr) λx r. Noe ha in his case T s = I sr and Ts 1 = I + sr. Thus (12) yields dν T dν (x) = exp X r (x +(1 s)r)ds and we obain (4). = exp X r (x + sr)ds References [B.1] D. Bell, A quasi-invariance heorem for measures on Banach spaces. Trans Amer. Mah. Soc. 29 (1985), [B.2] D. Bell, The Malliavin Calculus. Piman Monographs and Surveys in Pure and Applied Mahemaics, Vol. 34, Wiley, New York/ Longman U.K., [G-T] B. Gaveau and P. Trauber, L inegrale sochasique comme operaeur de divergence dans l espace foncionnel. J. Func. Anal. 46 (1982), [G] L. Gross, Absrac Wiener spaces, Proc. Fifh Berkeley Sympos. Mah. Sais. and Prob. vol. 2, par (i), (1965) [K.1] H. -H. Kuo, Inegraion on infinie-dimensional manifolds. Trans Amer. Mah. Soc. 159 (1971), [K.2] H. -H. Kuo, Gaussian Measures in Banach Spaces, Springer LNM No. 463, Springer-Verlag, Berlin, [7] R. Ramer, On nonlinear ransformaions of Gaussian measures. J. Func. Anal. 15 (1974),