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A CHANGE OF SCALE FORMULA FOR WIENER INTEGRALS OF CYLINDER FUNCTIONS ON
THE ABSTRACT WIENER SPACE II
YOUNG SIK KIM (Received 24 January 2000)
Abstract.We show that for certain bounded cylinder functions of the form F(x) = ˆ
µ((h1,x)∼,...,(hn,x)∼), x∈B, where ˆµ:Rn→Cis the Fourier-transform of the complex- valued Borel measureµonᏮ(Rn), the Borelσ-algebra ofRnwithµ<∞, the analytic Feynman integral ofFexists, although the analytic Feynman integral, limz→−iqIaw(F;z)= limz→−iq(z/2π)n/2
Rnf (→u)exp{−(z/2)| →u| 2}d→u, do not always exist for bounded cylin- der functionsF(x)=f ((h1,x)∼,...,(hn,x)∼), x∈B. We prove a change of scale formula for Wiener integrals ofFon the abstract Wiener space.
2000 Mathematics Subject Classification. Primary 28C20.
1. Introduction. In [3], Kim showed that for F ∈ Ᏺ(n;p), 1≤ p ≤ ∞, the ana- lytic Wiener integral exists and for F ∈Ᏺ(n;1), the analytic Feynman integral ex- ists and can be expressed as the limit of Wiener integrals and later he proved the change of scale formula for Wiener integrals for F ∈ Ᏺ(n;p), 1 ≤p ≤ ∞, where for 1 ≤p < ∞, Ᏺ(n;p) is the class of cylinder functions F of the form F(x) = f ((h1,x)∼,...,(hn,x)∼)andf:Rn→Ris inLp(Rn), andᏲ(n;∞)is the class of such cylinder functionsF, wheref:Rn→Ris inC0(Rn), the space of bounded continu- ous functions onRnthat vanish at infinity. But for 1< p≤ ∞, the analytic Feynman integral ofF ∈Ᏺ(n;p)donot always exist even ifF(x) =f ((h1,x)∼,...,(hn,x)∼) is a bounded cylinder function, as we cannot apply the Lebesgue dominated con- vergence theorem to the limit whenever z→ −iq through C+; limz→−iqIaw(F;z)= limz→−iq(z/2π)n/2
Rnf (→u)exp{−(z/2)| →u| 2}d→u.
In this paper, we show that the analytic Feynman integral ofF exists for certain bounded cylinder functions of the formF(x)=µ((hˆ 1,x)∼,...,(hn,x)∼), x∈B, where ˆ
µ:Rn→Cis the Fourier-transform of the complex-valued Borel measureµonᏮ(Rn), the Borelσ-algebra ofRnwithµ<∞. We establish the relationships between ana- lytic Wiener integrals, and analytic Feynman integrals, and we show that the analytic Feynman integral ofFcan be expressed as the limit of a sequence of Wiener integrals.
Later, we prove a change of scale formula for Wiener integrals ofF on the abstract Wiener space.
2. Definitions. LetH be a real separable infinite-dimensional Hilbert space with inner product·,·and norm· =
·,·. Let·0be a measurable norm onHwith respect tothe Gauss measureµ. LetB denote the completion ofH with respect to
·0. Letidenote the natural injection fromHintoB. The adjoint operatori∗ofiis
one-to-one and mapsB∗ continuously onto a dense subset ofH∗, whereH∗ andB∗ are topological duals ofHandB, respectively. By identifyingHwithH∗andB∗with i∗B∗, we have a triplet(B∗,H,B)such thatB∗⊂H∗≡H⊂Bandh,x =(h,x)for all hinB∗ andx inH, where(·,·)denotes the natural dual pairing betweenB∗andB.
By a well-known result of Gross [3],µ·i−1has a unique countably additive extension mtothe Borelσ-algebraᏮ(B)onB. The triplet(B,H,m)is called an abstract Wiener space andmis called a Wiener measure. We denote the Wiener integral of a functional F by
BF(x)dm(x). For more details see [1, 3].
Let{ej}∞j=1 denote a complete orthonormal system inH such thatej’s are inB∗. For eachh∈Handx∈B, we define a stochastic inner product(·,·)∼betweenHand Bas follows:
(h,x)∼=
n→∞lim n j=1
h,ej ej,x
, if the limit exists,
0, otherwise.
(2.1)
It is well known [2] that for everyh∈H,(h,x)∼ exists form-a.e.xinBand it has a Gaussian distribution with mean zero and variance|h|2. Furthermore, it is easy to show that(h,x)∼=(h,x)form-a.e.xinBifh∈B∗,(h,x)∼is essentially independent of the complete orthonormal set used in its definition, and finally we show that if {h1,...,hk} is an orthonormal set of elements in H, then (h1,x)∼,...,(hk,x)∼ are independent Gaussian functionals with mean zero and variance one. Note that if both handxare inH, then(h,x)∼= h,x.
Throughout this paper, letRn denote then-dimensional Euclidean space and let C,C+, andC∼+denote the complex numbers, the complex numbers with positive real part, and the nonzero complex numbers with nonnegative real part, respectively.
Definition2.1. Let(B,H,m)be an abstract Wiener space. A functionF is called acylinder functiononBif there exists a finite subset{g1,...,gk}ofHsuch that
F(x)=ψ g1,x∼
,...,
gk,x∼
, x∈B, (2.2)
where ψ is a complex-valued Borel measurable function onRk. It is easy toshow that there exists a linearly independent set{h1,...,hn}ofHsuch that every cylinder functionF of the form (2.2) is expressed as
F(x)=f h1,x∼
,...,
hn,x∼
, x∈B, (2.3)
wheref is a complex-valued Borel measurable function onRn. Thus we lose no gen- erality in assuming that every cylinder function onBis of the form (2.3).
Definition2.2. LetFbe a complex-valued measurable function onBsuch that the integral
J(F;λ)=
BF λ−1/2x
dm(x) (2.4)
exists for all realλ >0. If there exists a functionJ∗(F;z)analytic onC+ such that
J∗(F;λ)=J(F;λ)for all realλ >0, then we defineJ∗(F;z)tobe theanalytic Wiener integralofF overBwith parameterz, and for eachz∈C+, we write
Iaw(F;z)=J∗(F;z). (2.5)
Letqbe a nonzero real number and letF be a function onBwhose analytic Wiener integral exists for eachzinC+. If the following limit exists, then we call it theanalytic Feynman integralofFoverBwith parameterq, and we write
Iaf(F;q)= lim
z→−iqIaw(F;z), (2.6)
wherezapproaches−iqthroughC+andi2= −1.
Definition2.3. Let ᏹ(Rn)denote the space of complex-valued Borel measures onᏮ(Rn), the Borel σ-algebra ofRn. It is well known that a complex-valued Borel measureµnecessarily has a finite total variationµ, andᏹ(Rn)is a Banach algebra under the norm·and with convolution as multiplication.
Let µ be in ᏹ(Rn). Then the Fourier transformation µˆof µ is a complex-valued function defined onRnby the formula
ˆ µ
# u
=
Rnexp
i→u, →v µ d→v
, →u ∈Rn, (2.7)
where→u =(u1,...,un)and→v =(v1,...,vn)are inRn, and→u, →v =n
j=1ujvj. We will close this section by mentioning the following useful theorem which is called the Wiener integration formula.
Theorem2.4. Let(B,H,m)be an abstract Wiener space and let{h1,...,hn}be an orthonormal set of elements inH. LetF:B→Cbe a function defined by the formula
F(x)=f h1,x∼
,...,
hn,x∼
, x∈B, (2.8)
wheref:Rn→Cis a Lebesgue measurable function. Then
Bf h1,x∼
,...,
hn,x∼
dm(x)= 1
2π n/2
Rnf→u exp
−1 2→u 2
d→u, (2.9) where→u =(u1,...,un)∈Rn, |→u| 2=n
j=1u2j, andd→u =du1···dun.
In the next section, we use several times the following well-known integration for-
mula
Rexp
−au2+ibu du=
π aexp
−b2 4a
, (2.10)
whereais a complex number with Rea >0,bis a real number, andi2= −1.
3. The main results. In this paper, we give a class of a certain bounded cylinder functions of the formF(x)=f ((h1,x)∼,...,(hn,x)∼),x∈B, such thatf :Rn→C
is bounded, whose analytic Wiener and analytic Feynman integral ofF exist and we show that the analytic Feynman integral of such cylinder functions can be expressed as the limit of Wiener integrals. Later, we prove a change of scale formula for Wiener integrals of such cylinder functions on the abstract Wiener space.
Define the functionF:B→Cby F(x)=µˆ
h1,x∼ ,...,
hn,x∼
, x∈B, (3.1)
where ˆµis the Fourier transform of complex-valued Borel measuresµinᏹ(Rn)and ᏹ(Rn) is as in Definition 2.3. Then F : B → C is a bounded cylinder function, as
|ˆµ(#u)| ≤ µ<∞.
First, we show that the analytic Wiener and the analytic Feynman integrals of the functionF exist.
Theorem3.1. Let(B,H,m)be an abstract Wiener space and let{h1,...,hn}be an orthonormal set of elements in H. LetF :B→Cbe given by (3.1). Then the analytic Wiener and the analytic Feynman integrals ofFexist, and for everyz∈C+,
Iaw(F;z)=
Rnexp
− 1 2z→v 2
µ d→v
(3.2) and for every nonzero real numberq,
Iaf(F;q)=
Rnexp
− i 2q→v 2
µ d→v
, (3.3)
where→v =(v1,...,vn)∈Rnand|→v |2=n
j=1vj2.
Proof. By Fubini’s theorem and by Theorem 2.4 and by (2.10), we have that for all positive real numberλ,
J(F:λ)≡
BF λ−1/2x
dm(x)=
Bµˆ
h1,λ−1/2x∼ ,...,
hn,λ−1/2x∼ dm(x)
=
Rn
Bexp
iλ−1/2 n j=1
vj
hj,x∼
dm(x)dµ v#
= 1 2π
n/2
Rn
Rnexp
iλ−1/2 n j=1
vjuj
exp
−1 2
n j=1
u2j
du1···dundµ
# v
=
Rnexp
− 1 2λ
n j=1
vj2
dµ
# v
.
(3.4) Now let J∗(F : z)=
Rnexp
−(1/2z)n
j=1vj2
dµ(#v), z∈C+. Then J∗(F :λ)= J(F:λ)for all realλ >0. By dominated convergence theorem,J∗(F:z)is continuous onC+. Since exp
−(1/2z)n
j=1vj2
is analytic onC+for eachv#=(v1,...,vn)∈Rn, we have that
Γexp
−(1/2z)n
j=1vj2
dz=0 for all rectifiable simple closed curve Γ lyingC+ by Cauchy integral theorem. Asexp
−(1/2z)n
j=1vj2'1 for allz∈ C+,we can apply Fubini’s theorem to the integral
ΓJ∗(F :z)dzand then we have
ΓJ∗(F:z)dz=0.By Morera’s theorem,J∗(F:z)is an analytic function ofzinC+.
Therefore the analytic Wiener integralIaw(F :z) exist and we have (3.2). Toprove (3.3), letfn(#v)=exp{−(1/zn)|#v|2}, zn∈C+and letzn→ −iqwhenevern→ ∞. Then fn(#v)→f (#v)≡exp−{(i/2q)#v2}, wheneverzn→ −iqand|fn(#v)| ≤1, for allzn∈C+. By the bounded convergence theorem, we have (3.3), asµ(Rn)<∞.
In order to obtain our main results, we need the following lemma.
Lemma3.2. Let(B,H,m)be an abstract Wiener space and let {h1,...,hn}be an orthonormal set of elements inH. LetF:B→Cbe given by (3.1). Then for everyz∈C+, the functional
exp 1−z
2 n j=1
hj,x∼2
F(x) (3.5)
is Wiener integrable onB.
Proof. By Theorem 2.4, we have that for everyz∈C+,
Bexp 1−z
2 n j=1
hj,x∼2
F(x)dm(x)= 1 2π
n/2
Rnµˆ→u exp
−z 2→u 2
d→u, (3.6) where →u =(u1,...,un)∈ Rn, |→u| 2 =n
j=1u2j, and d→u =du1···dun. Because the absolute value of the last integral is less thanµ·z−n/2, the proof of this lemma is complete.
Theorem3.3. Let(B,H,m)be an abstract Wiener space and let{h1,...,hn}be as in Definition 2.3. LetF :B→Cbe given by (3.1). Then for everyz∈C+, the analytic Wiener integralIaw(F;z)ofF is expressed as follows:
Iaw(F;z)=zn/2
Bexp
(1−z) 2
n j=1
hj,x∼2
F(x)dm(x). (3.7) Proof. By Lemma 3.2, the right-hand side of (3.7) has a finite value. Now let us calculate the following Wiener integral:
Bexp
(1−z) 2
n j=1
hj,x∼2
F(x)dm(x)
= 1 2π
n/2
Rnµˆ→u exp
−z 2→u 2
d→u
= 1
2π n/2
Rn
Rnexp
i n j=1
vjuj
µ
d→v exp
−z 2→u 2
d→u
= 1
2π n/2
Rn
n
j=1
Rexp
−z
2u2j+ivjuj
duj
µ
d→v
=z−n/2
Rnexp
− 1 2z→v 2
µ d→v
.
(3.8)
Here, the first equality comes from Theorem 2.4, the second equality comes from the definition of Fourier transform ˆµ ofµ∈ᏹ(Rn), the third equality follows from Fubini’s theorem, and the last equality follows from the formula (2.10). From (3.2) and (3.8), we have the desired result (3.7).
Now we express the analytic Feynman integralIaf(F;q)ofFas the limit of a sequence of Wiener integrals on the abstract Wiener space.
Theorem3.4. Let(B,H,m)be an abstract Wiener space and let{h1,...,hn}be an orthonormal set of elements in H. LetF:B→Cbe given by (3.1). If{zk}∞k=1is a sequence of complex numbers fromC+ such thatzkapproaches−iqthroughC+, whereqis a nonzero real number andi2= −1, then the analytic Feynman integralIaf(F;q)ofF is expressed as follows:
Iaf(F;q)=lim
k→∞
zkn/2
Bexp 1−zk
2 n j=1
hj,x∼2
F(x)dm(x). (3.9) Proof. We can obtain from (3.8) that
zn/2k
Bexp 1−zk 2
n j=1
hj,x∼2
F(x)dm(x)=
Rnexp
− 1 2zk
→v 2 µ
d→v . (3.10) Lettingk→ ∞in (3.10) and using the dominated convergence theorem, we have
k→∞lim
zkn/2
Bexp 1−zk
2 n j=1
hj,x∼2
F(x)dm(x)
=lim
k→∞
Rnexp
− 1 2zk
→v 2 µ
d→v
=
Rnexp
− i 2q→v 2
µ d→v
.
(3.11)
From (3.2) and (3.11), (3.9) follows immediately.
Finally, we obtain a change of scale formula for Wiener integrals forF:B→Cwhich was given by (3.1).
Theorem3.5. Let(B,H,m)be an abstract Wiener space. Letρ >0be given and let
h1,...,hn
be an orthonormal set of elements in H. Then forF:B→Cwhich was given by (3.1),
BF(ρx)dm(x)=ρ−n
Bexp ρ2−1
2ρ2 n j=1
hj,x∼2
F(x)dm(x). (3.12) Proof. First, we know that for all realz >0,Iaw(F;z)=
BF(z−1/2x)dm(x)by Definition 2.2. Using Theorem 3.3 and takingz=ρ−2in the above equality, we have the desired result.
Acknowledgement. This paper was supported by BK-21 Project.
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Young Sik Kim: BK-21Mathematical Science Division, Department of Mathematics, Seoul National University, Seoul,151-742, Korea
