Conditional paths in abstract Wiener space : an ${rm L}_p$ theory

CONDITIONAL FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION PRODUCT OVER WIENER PATHS IN

ABSTRACT WIENER SPACE : AN L _p THEORY

Dong Hyun Cho

Abstract. In this paper, using a simple formula, we evaluate the conditional Fourier-Feynman transforms and the conditional con- volution products of cylinder type functions, and show that the conditional Fourier-Feynman transform of the conditional convolu- tion product is expressed as a product of the conditional Fourier- Feynman transforms. Also, we evaluate the conditional Fourier- Feynman transforms of the functions of the forms

exp

Z

T 0

θ(s, x(s))ds



, exp

Z

T 0

θ(s, x(s))ds



φ(x(T )),

exp

Z

T 0

θ(s, x(s))dζ(s)



, exp

Z

T 0

θ(s, x(s))dζ(s)

 φ(x(T )) which are of interest in Feynman integration theories and quantum mechanics.

1. Introduction and preliminaries

Let C ₀ [0, T ] denote the classical Wiener space, that is, the space of real-valued continuous functions x(t) on [0, T ] with x(0) = 0. The con- cept of conditional Wiener integral on Wiener space was introduced by Yeh in [18, 19]. By a conditional Wiener integral we mean the condi- tional expectation E[F |X] of a real or complex-valued Wiener integrable function F conditioned by a Wiener measurable function X on C ₀ [0, T ],

Received January 5, 2003. Revised March 12, 2003.

2000 Mathematics Subject Classification: 28C20.

Key words and phrases: conditional analytic Feynman integral, conditional convo- lution product, conditional Fourier-Feynman transform, conditional Wiener integral, cylinder type function, simple formula for conditional Wiener integral.

Research supported by the Basic Science Research Institute Program, Korea Re-

search Foundation under Grant KRF 2001-005-D20004.

which is given as a function on the value space of X. We shall be con- cerned exclusively with X given by X(x) = (x(t ₁ ), . . . , x(t _k )), where 0 < t ₁ < · · · < t ^k = T for any fixed positive integer k. In [14], Park and Skoug derived a simple formula for the conditional Wiener integral with the conditioning function X. They then used this formula to express conditional Wiener integrals directly in terms of ordinary Wiener inte- grals. In [15], Park and Skoug introduced the concept of a conditional Fourier-Feynman transform and the concept of a conditional convolu- tion product and obtained several formulas relating these concepts; in particular see equations (3.13) and (4.11) in [15]. In [5], using ideas and formulas developed in [15], Chang and Skoug examined the effects that drift had on conditional Fourier-Feynman transforms and conditional convolution products. In section 4 of [5] and in section 4 of [15] the au- thors established various formulas involving conditional transforms and conditional convolution products for functions in the Banach algebra S which was introduced by Cameron and Storvick in [1].

In [12], Kuelbs and Lepage introduced C 0 (B), the space of continuous functions on [0, T ] into B which vanish at 0. In [16], Ryu established various properties involving C 0 (B). In [20], Yoo introduced the Banach algebra S _B ^′′ which corresponds to Cameron and Storvick’s space S ^′′ in [1]. Chang and his coworkers introduced the class A ^{(p) n,s} (1 ≤ p ≤ ∞) of cylinder type functions defined on C ₀ (B), and then established rela- tionships between the Fourier-Feynman transform and the convolution product of functions in A ^{(p) n,s} ([4]).

In [2], Chang and his coworkers defined the L ₁ conditional Fourier- Feynman transform and the conditional convolution product on the space C ₀ (B), and they investigate relationships between them.

In this paper, we define the L p conditional Fourier-Feynman trans- form on C ₀ (B) for 1 < p ≤ ∞, and we evaluate the L ^p conditional Fourier-Feynman transforms and the conditional convolution products of functions in A ^{(p) n,s} , and then, we show that the conditional Fourier- Feynman transform of the conditional convolution product can be ex- pressed as a product of the conditional Fourier-Feynman transforms for these functions. Also, for 1 ≤ p ≤ ∞, we evaluate the L ^p conditional Fourier-Feynman transforms of the functions of the forms

exp

Z T 0

θ(s, x(s))ds



, exp

Z T 0

θ(s, x(s))ds



φ(x(T )), exp

Z T 0

θ(s, x(s))dζ(s)



, exp

Z T 0

θ(s, x(s))dζ(s)



φ(x(T ))

which are of interest in Feynman integration theories and quantum me- chanics.

Let (Ω, A, P ) be a probability space, let B be a real normed linear space with norm k · k and let B(B) be the Borel σ-field on B. Let X : (Ω, A, P ) → (B, B(B)) be a random variable and let F : Ω → C be an integrable function. Let P X be the probability distribution of X on (B, B(B)) and let D be the σ-field {X ⁻¹ (A) : A ∈ B(B)}. Let P D be the probability measure induced by P , that is, P _D (E) = P (E) for E ∈ D.

By the definition of conditional expectation there exists a D-measurable function E[F |X](the conditional expectation of F given X) defined on Ω such that the relation

Z

E

E [F |X](ω) dP D (ω) = Z

E

F (ω) dP (ω)

holds for every E ∈ D. But there exists a P X -integrable function ψ defined on B which is unique up to P X -a.e. such that E[F |X](ω) = (ψ◦X)(ω) for P D -a.e. ω in Ω. ψ is also called the conditional expectation of F given X and without loss of generality, it is denoted by E[F |X](ξ) for ξ ∈ B. Throughout this paper, we will consider the function ψ as the conditional expectation of F given X.

2. Wiener paths in abstract Wiener space

Let (H, B, m) be an abstract Wiener space([13]). Let {e ^j : j ≥ 1} be a complete orthonormal set in the real separable Hilbert space H such that e _j ’s are in B ^∗ , the dual space of real separable Banach space B. For each h ∈ H and y ∈ B, define the stochastic inner product (h, y) ^∼ by

(h, y) ^∼ =

 lim _n→∞ P n

j=1 hh, e ^j i(y, e ^j ), if the limit exists;

0, otherwise,

where (·, ·) denotes the dual pairing between B and B ^∗ ([11]). Note that for each h(6= 0) in H, (h, ·) ^∼ is a Gaussian random variable on B with mean zero, variance |h| ² ; also (h, y) ^∼ is essentially independent of the choice of the complete orthonormal set used in its definition and further, (h, λy) ^∼ = (λh, y) ^∼ = λ(h, y) ^∼ for all λ ∈ R. It is well-known that if {h 1 , h ₂ , . . . , h n } is an orthogonal set in H, then the random variables (h j , ·) ^∼ ’s are independent. Moreover, if both h and y are in H, then (h, y) ^∼ = hh, yi.

Let C ₀ (B) denote the set of all continuous functions on [0, T ] into B

which vanish at 0. Then C ₀ (B) is a real separable Banach space with

the norm kxk C

₀

(B) ≡ sup 0≤t≤T kx(t)k B . The minimal σ-field making the

mapping x → x(t) measurable is B(C 0 (B)), the Borel σ-field on C ₀ (B).

Further, Brownian motion in B induces a probability measure m _B on (C ₀ (B), B(C 0 (B))) which is mean-zero Gaussian. We will find a concrete form of m _B . Let ~t = (t ₁ , t ₂ , . . . , t _k ) be given with 0 = t ₀ < t ₁ < t ₂ <

· · · < t ^k ≤ T . Let T ~ t : B ^k → B ^k be given by T _{~ t} (x ₁ , x ₂ , . . . , x _k )

=



(t ₁ − t 0 )

¹²

x ₁ , (t ₁ − t 0 )

¹²

x ₁ + (t ₂ − t 1 )

¹²

x ₂ , . . . ,

k

X

j=1

(t j − t j−1 )

¹²

x _j



 . We define a set function ν _{~ t} on B(B ^k ) by

ν _{~ t} (B) =

k

Y

1

m

!

 T _~ ⁻¹

t (B) 

for B ∈ B(B ^k ). Then ν _{~ t} is a Borel measure. Let f _{~ t} : C ₀ (B) → B ^k be the function defined by

f _{~ t} (x) = (x(t ₁ ), x(t ₂ ), . . . , x(t k )).

For Borel subsets B ₁ , B ₂ , . . . , B k of B, f _{~ t} ⁻¹ ( Q k

j=1 B j ) is called the I-set with respect to B 1 , B ₂ , . . . , B _k . Then the collection I of all I-sets is a semi-algebra. We define a set function m _B on I by

m _B



f _~ ⁻¹

t





k

Y

j=1

B _j







 = ν ~ t





k

Y

j=1

B _j



 .

Then m _B is well-defined and countably additive on I. Using Carath´eod- ory extension process, we have a Borel measure m _B on B(C ⁰ (B)).

Now, we introduce Wiener integration theorem without proof. For the proof see [16].

Theorem 1 (Wiener integration theorem). Let ~t = (t ₁ , t ₂ , . . . , t _k ) be given with 0 = t ₀ ≤ t 1 ≤ t 2 ≤ · · · ≤ t ^k ≤ T and let f : B ^k → C be a Borel measurable function. Then

Z

C

0

(B)

f(x(t 1 ), x(t 2 ), . . . , x(t k )) dm _B (x)

= ∗

Z

B

^k

(f ◦ T ~ t )(x ₁ , x ₂ , . . . , x _k ) d





k

Y

j=1

m



 (x ₁ , x ₂ , . . . , x _k ),

where by = we mean that if either side exists, then both sides exist and ^∗ they are equal.

Let τ : 0 = t ₀ < t ₁ < · · · < t ^k = T be a partition of [0, T ] and let x be in C ₀ (B). Define the polygonal function [x] of x on [0, T ] by

(1) [x](t) =

k

X

j=1

χ _(t

_j−1

_,t

_j

_] (t)



x(t _j−1 ) + t − t j−1

t _j − t ^j−1 (x(t j ) − x(t j−1 ))

 , where t ∈ [0, T ]. For each ~ξ = (ξ 1 , . . . , ξ _k ) ∈ B ^k , let [~ ξ] be the polygonal function of ~ ξ on [0, T ] given by (1) with replacing x(t j ) by ξ j for j = 0, 1, . . . , k (ξ ₀ = 0).

The following lemmas are useful to the next sections. For the detailed proof, see [3].

Lemma 2. If {x(t) : 0 ≤ t ≤ T } is the Wiener process on C ⁰ (B) × [0, T ], then {x(t) − [x](t) : t j−1 ≤ t ≤ t ^j }, where j = 1, . . . , k, are stochastically independent.

Lemma 3. Let F be defined and integrable on C ₀ (B). Let X _τ : C ₀ (B) → B ^k be a random variable given by X τ (x) = (x(t ₁ ), . . . , x(t _k )).

Then for every Borel measurable subset B of B ^k , µ τ (B) ≡

Z

X

τ⁻¹

(B)

F (x) dm _B (x) = Z

B

E [F (x − [x] + [~ξ])] dP ^X

τ

(~ ξ), where P _X

_τ

is the probability distribution of X _τ on (B ^k , B(B ^k )).

By the definition of conditional expectation and Lemma 3, we have E [F |X ^τ ](~ ξ ) = E[F (x − [x] + [~ξ])] for P X

τ

-a.e. ~ ξ.

(2)

The function E[F |X ^τ ] is called the conditional Wiener integral of F given X τ . Note that, for the definition of the conditional Wiener integral, X τ

may be any random variable defined on C 0 (B). The equation (2) is called a simple formula for conditional Wiener integral on the space C ₀ (B).

For λ > 0 and ~ ξ ∈ B ^k , suppose E[F (λ ⁻

¹²

·)|X ^τ (λ ⁻

¹²

·)](~ξ) exists. From (2) we have

E[F (λ ⁻

¹²

·)|X ^τ (λ ⁻

¹²

·)](~ξ) = E[F (λ ⁻

¹²

(x − [x]) + [~ξ])]

for a.e. ~ ξ ∈ B ^k . If, for ~ ξ ∈ B ^k , E[F (λ ⁻

¹²

(x − [x]) + [~ξ])] has the analytic extension J λ (~ ξ) on C ₊ ≡ {λ ∈ C : Re λ > 0}, then we write

J _λ (~ ξ) = E ^anw

^λ

[F |X ^τ ](~ ξ)

for λ ∈ C + . In this case, we call J λ (~ ξ) a version of conditional analytic Wiener integral of F given X τ .

For non-zero real q and ~ ξ ∈ B ^k , if the limit

λ→−iq lim E ^anw

^λ

[F |X ^τ ](~ ξ)

exists, where λ approaches to −iq through C + , then we write

λ→−iq lim E ^anw

^λ

[F |X ^τ ](~ ξ) = E ^anf

^q

[F |X ^τ ](~ ξ).

In this case, we call E ^anf

^q

[F |X ^τ ](~ ξ) a version of conditional analytic Feynman integral of F given X τ .

3. Conditional Fourier-Feynman transform of cylinder type functions

In this section, we define L p (1 < p ≤ ∞) conditional Fourier-Feynman transform over Wiener paths in abstract Wiener space and evaluate them for cylinder type functions. We also evaluate conditional convolution products for these type of functions.

A subset E of C ₀ (B) is called a scale-invariant null set if m _B (λE) = 0 for any λ > 0 and a property is said to hold s-a.e. if it holds except for a scale-invariant null set.

For a given real number p with 1 < p ≤ ∞, suppose p and p ^′ are related by ¹ _p + _p ¹

′

= 1 (possibly p ^′ = 1 if p = ∞). Let G ⁿ and G be measurable functions such that, for each γ > 0,

n→∞ lim Z

C

0

(B) |G n (γx) − G(γx)| ^p

^′

dm _B (x) = 0.

Then we write

l.i.m.

n→∞ (w _s ^p

^′

)(G n ) ≈ G

and call G the scale-invariant limit in the mean of order p ^′ . A similar definition is understood when n is replaced by a continuously varying parameter.

Let H be a real separable infinite dimensional Hilbert space, let n be a positive integer and let {h 1 , . . . , h n } be an orthonormal set in H. For 1 ≤ p < ∞ let A ^{(p) n,s} be the space of all cylinder type functions F defined on C 0 (B) of the form

(3) F (x) = f ((h 1 , x(s)) ^∼ , . . . , (h n , x(s)) ^∼ )

for s-a.e. x in C ₀ (B) where f : R ⁿ → R is in L ^p (R ⁿ ) and s ∈ (0, T ].

Let A ^{(0) n,s} be the space of all functions of the form (3) with f ∈ C 0 (R ⁿ ), the space of continuous functions on R ⁿ which vanish at infinity and let A ^{(∞) n,s} be the space of all functions of the form (3) with f ∈ L ^∞ (R ⁿ ), the space of essentially bounded functions on R ⁿ . Note that, without loss of generality, we can take f to be Borel measurable.

For convenience, we let

Γ = t _p

∗

− t ^p

^∗

−1

(t p

^∗

− s)(s − t ^p

^∗

−1 ) (4)

for t p

^∗

−1 < s < t _p

∗

and φ _λ (~u) =  λ 2π



ⁿ₂

exp



− λ 2

n

X

j=1

u ² _j



(λ

¹²

∈ C + ) (5)

where ~u = (u ₁ , . . . , u n ) ∈ R ⁿ and λ ∈ C ^∼ + ≡ {λ ∈ C : Re λ ≥ 0} − {0}.

Also, we let for ~ ξ ∈ B ^k

~

w _{ξ ~} = (w _{~ ξ1} , . . . , w _{ξn ~} ) = ((h ₁ , [~ ξ](s)) ^∼ , . . . , (h _n , [~ ξ](s)) ^∼ ) (6)

and

~

w _y = (w _y1 , . . . , w _yn ) = ((h ₁ , y(s)) ^∼ , . . . , (h _n , y(s)) ^∼ ) (7)

for y in C 0 (B).

Definition 4. Let F be defined on C ₀ (B) and let X τ be given as in Lemma 3. For λ ∈ C ⁺ and for s-a.e. ~ ξ ∈ B ^k let

T _λ [F |X τ ](y, ~ ξ) = E ^anw

^λ

[F (y + ·)|X τ ](~ ξ)

for s-a.e. y ∈ C ⁰ (B) if it exists. For non-zero real q and for s-a.e. ~ ξ ∈ B ^k , we define the L ₁ conditional Fourier-Feynman transform T q ⁽¹⁾ [F |X ^τ ] of F given X τ by the formula

T _q ⁽¹⁾ [F |X ^τ ](y, ~ ξ) = lim

λ→−iq T _λ [F |X ^τ ](y, ~ ξ)

if it exists for s-a.e. y ∈ C 0 (B) and for 1 < p ≤ ∞ we define the L ^p conditional Fourier-Feynman transform T q ^(p) [F |X ^τ ] of F given X τ by the formula

T _q ^(p) [F |X τ ](·, ~ξ) ≈ l.i.m.

λ→−iq (w ^p _s

^′

)(T _λ [F |X τ ](·, ~ξ)),

where λ approaches to −iq through C + .

Theorem 5. Let F ∈ A ^{(p) n,s} (1 ≤ p ≤ ∞) be given by (3). Let X ^τ be given as in Lemma 3 and let t p

^∗

−1 < s ≤ t ^p

^∗

for some p ^∗ ∈ {1, . . . , k}.

Then, for any λ ∈ C + and for s-a.e. ~ ξ ∈ B ^k , T _λ [F |X τ ](y, ~ ξ) exists for s-a.e. y ∈ C ⁰ (B) and T λ [F |X ^τ ](·, ~ξ) ∈ A ^{(p) n,s} . Moreover, when t _p

∗

−1 <

s < t p

^∗

, we have

T _λ [F |X ^τ ](y, ~ ξ) = (φ _λΓ ∗ f)( ~ w y + ~ w _{ξ ~} ) (8)

and, when s = t _p

∗

, we have

T _λ [F |X ^τ ](y, ~ ξ) = F (y + [~ ξ]) = f ( ~ w _y + ~ w _{~ ξ} ), (9)

where Γ, φ _λΓ , w ~ _{~ ξ} and w ~ _y are given by (4), (5), (6) and (7), respectively.

Proof. Let t p

^∗

−1 < s < t p

^∗

and let λ > 0. Using the same method in the proof of Theorem 4.2 in [2], we have

E[F (y + λ ⁻

¹²

(x − [x]) + [~ξ])] = (φ λΓ ∗ f)( ~ w _y + ~ w _{~ ξ} )

for s-a.e. ~ ξ ∈ B ^k and for s-a.e. y ∈ C 0 (B). By Morera’s theorem, we have (8). The fact T λ [F |X ^τ ](·, ~ξ) ∈ A ^{(p) n,s} follows from Young’s inequality in [7, p.232].

When s = t p

^∗

, the equation (9) follows easily since E[F (y + λ ⁻

¹²

(x − [x]) +~[ξ])] = F (y + [~ ξ]) for λ > 0.

Theorem 6. For 1 ≤ p ≤ 2 let F ∈ A ^{(p) n,s} be given by (3) and let

1

p + _p ¹

_′

= 1(p ^′ = ∞ if p = 1). Let X ^τ be given as in Lemma 3 and let t _p

∗

−1 < s ≤ t ^p

^∗

for some p ^∗ ∈ {1, . . . , k}. Then, for any non-zero real q, and for s-a.e. ~ ξ ∈ B ^k , T q ^(p) [F |X ^τ ](y, ~ ξ ) exists for s-a.e. y ∈ C ⁰ (B).

Moreover, when t p

^∗

−1 < s < t p

^∗

, we have

T _q ^(p) [F |X ^τ ](y, ~ ξ) = (φ _−iqΓ ∗ f)( ~ w y + ~ w _{ξ ~} )

with T q ^(p) [F |X ^τ ](·, ~ξ) ∈ A ^{(p n,s}

^′

⁾ (in fact, T q ⁽¹⁾ [F |X ^τ ](·, ~ξ) ∈ A ^{(0) n,s} ) and, when s = t _p

∗

, we have

T _q ^(p) [F |X ^τ ](y, ~ ξ) = F (y + [~ ξ]) = f ( ~ w _y + ~ w _{ξ ~} ) (10)

with T q ^(p) [F |X ^τ ](·, ~ξ) ∈ A ^{(p) n,s} , where Γ, φ _−iqΓ , w ~ _{~ ξ} and w ~ y are given by (4), (5), (6) and (7), respectively.

Proof. When p = 1, the result follows from Theorem 4.2 in [2].

Let 1 < p ≤ 2, t ^p

^∗

−1 < s < t p

^∗

. Then, for γ > 0, for s-a.e. ~ ξ in B ^k and by Theorems 1, 5, we have

λ→−iq lim Z

C

₀

(B) |(φ λΓ ∗ f)(γ ~ w _y + ~ w _{~ ξ} ) − (φ −iqΓ ∗ f)(γ ~ w _y + ~ w _{ξ ~} )| ^p

^′

dm _B (y)

= lim

λ→−iq

Z

B |(φ λΓ ∗ f)(γ √

s((h ₁ , x ₁ ) ^∼ , . . . , (h n , x ₁ ) ^∼ ) + ~ w _{~ ξ} )

−(φ −iqΓ ∗ f)(γ √

s((h ₁ , x ₁ ) ^∼ , . . . , (h _n , x ₁ ) ^∼ ) + ~ w _{~ ξ} )| ^p

^′

dm(x ₁ )

≤ lim

λ→−iq

 1

2πsγ ²



ⁿ

2

k(φ λΓ ∗ f)(· + ~ w _{ξ ~} ) − (φ −iqΓ ∗ f)(· + ~ w _{~ ξ} )k ^p p

^′′

which converges to 0 as λ approaches to −iq through C ⁺ , by Lemmas 1.1, 1.2 in [10] and since (h j , ·) ^∼ is normally distributed with mean 0, variance 1.

When s = t p

^∗

, the result follows trivially by Theorem 5.

Remark 1. (10) holds for 1 ≤ p ≤ ∞.

Next we establish an inverse conditional Fourier-Feynman transform theorem for functions in A ^{(p) n,s} .

Theorem 7. For 1 ≤ p < ∞ let F ∈ A ^{(p) n,s} be given by (3) and let X τ be given as in Lemma 3. Let q be a non-zero real number and let w ~ _y be given by (7). For (~ ξ ₁ , ~ ξ ₂ ) ∈ B ^k × B ^k and for y ∈ C 0 (B), let F _{~ ξ}

1

,~ ξ

2

(y) = f ( ~ w y + ~ w ₁ + ~ w ₂ ), where ~ w ₁ = (w ₁₁ , . . . , w _1n ) = ((h ₁ , [~ ξ ₁ ](s)) ^∼ , . . . , (h n , [~ ξ ₁ ](s)) ^∼ ), ~ w ₂ = (w ₂₁ , . . . , w _2n ) = ((h ₁ , [~ ξ ₂ ](s)) ^∼ , . . . , (h n , [~ ξ ₂ ](s)) ^∼ ). Then we have

(i) for s-a.e. (~ ξ ₁ , ~ ξ ₂ ) ∈ B ^k × B ^k and for γ > 0, we have

λ→−iq lim Z

C

₀

(B) |T λ [T λ [F |X ^τ ](·, ~ ξ ₁ )|X ^τ ](γy, ~ ξ ₂ ) − F _ξ ^~

₁

_{,~ ξ}

₂

(γy)| ^p dm _B (y) = 0 and

(ii) for s-a.e. (~ ξ ₁ , ~ ξ ₂ ) ∈ B ^k × B ^k and for s-a.e. y ∈ C 0 (B), we have T _λ [T _λ [F |X τ ](·, ~ ξ ₁ )|X τ ](y, ~ ξ ₂ ) −→ F ~ ξ

1

,~ ξ

2

(y),

where λ approaches to −iq through C + .

Proof. Let t _p

∗

−1 < s < t _p

∗

for some p ^∗ ∈ {1, . . . , k} and let ~u =

(u 1 , . . . , u _n ) ∈ R ⁿ . Then for λ ∈ C ⁺ , for s-a.e. (~ ξ ₁ , ~ ξ ₂ ) ∈ B ^k × B ^k and

for s-a.e. y ∈ C 0 (B), using the same method in the proof of Theorem 4.4 in [2], we have

T _λ [T λ [F |X ^τ ](·, ~ ξ ₁ )|X ^τ ](y, ~ ξ ₂ )

=  |λ| ² Γ 4πReλ



ⁿ

2

Z

R

ⁿ

f (~u) exp



− |λ| ² Γ 4Reλ

n

X

j=1

(u j − w ^yj − w 1j − w 2j ) ²

 d~ u where Γ is given by (4). Let γ > 0 and for ~v = (v ₁ , . . . , v _n ) ∈ R ⁿ let

k _{~ ξ}

1

,~ ξ

2

(λ, ~v)

=  |λ| ² Γ 4πReλ



ⁿ₂

Z

R

ⁿ

f (~u) exp



− |λ| ² Γ 4Reλ

n

X

j=1

(u _j − v j − w 1j − w 2j ) ²

 d~ u.

Then, by Wiener integration theorem(Theorem 1) and the change of variable theorem, we have

Z

C

0

(B) |T λ [T _λ [F |X ^τ ](·, ~ξ 1 )|X ^τ ](γy, ~ ξ ₂ ) − F ~ ξ

1

,~ ξ

2

(γy)| ^p dm _B (y)

= Z

B |k ^~ ξ

₁

,~ ξ

₂

(λ, γ √

s((h 1 , x ₁ ) ^∼ , . . . , (h n , x ₁ ) ^∼ ))

−f(γ √

s((h 1 , x ₁ ) ^∼ , . . . , (h n , x ₁ ) ^∼ ) + ~ w ₁ + ~ w ₂ )| ^p dm(x 1 )

=

 1

2πsγ ²



ⁿ

2

Z

R

ⁿ

|k ~ ξ

1

,~ ξ

2

(λ, ~z) − f(~z + ~ w ₁ + ~ w ₂ )| ^p

× exp



− 1 2sγ ²

n

X

j=1

z _j ²

 d~ z

where ~z = (z ₁ , . . . , z _n ), since (h j , ·) ^∼ is normally distributed with mean 0 and variance 1. Let ǫ = ( _|λ| ^2Reλ

2

Γ )

¹²

and let ϕ ǫ (~u) = _ǫ ¹

n

φ ₁ ( ^{~ u} _ǫ ), where φ ₁ is given by (5). Then, R

R

ⁿ

φ ₁ (~u)d~u = 1 and hence in view of Theorem 1.18 in [17], we have

λ→−iq lim Z

C

₀

(B) |T λ [T λ [F |X ^τ ](·, ~ξ 1 )|X ^τ ](γy, ~ ξ ₂ ) − F ~ ξ

1

,~ ξ

2

(γy)| ^p dm _B (y)

≤ lim

λ→−iq

 1

2πsγ ²



ⁿ

2

k(f ∗ ϕ ^ǫ )(· + ~ w ₁ + ~ w ₂ ) − f(· + ~ w ₁ + ~ w ₂ )k ^p p

= 0

where λ approaches to −iq through C + . This proves (i). Also, (ii)

follows from Theorem 1.25 in [17].

Suppose that s = t p

^∗

for some p ^∗ ∈ {1, . . . , k} and let γ > 0. Then, for λ ∈ C + , for s-a.e. (~ ξ ₁ , ~ ξ ₂ ) ∈ B ^k × B ^k and for s-a.e. y ∈ C 0 (B), we have

T _λ [T λ [F |X ^τ ](·, ~ ξ ₁ )|X ^τ ](γy, ~ ξ ₂ ) = F ~ ξ

1

,~ ξ

2

(γy)

by Theorem 5. Then (i) follows trivially and (ii) follows when γ = 1.

Remark 2. (ii) of Theorem 7 holds for p = ∞.

4. Conditional convolution product and conditional trans- formation of conditional convolution product

Now we define conditional convolution product and investigate re- lationships between conditional convolution product and conditional Fourier Feynman transform.

Definition 8. Let X _τ be given as in Lemma 3 and let F, G be defined on C ₀ (B). We define the conditional convolution product [(F ∗ G) ^λ |X ^τ ] of F, G given X _τ by the formula, for s-a.e. ~ ξ ∈ B ^k

[(F ∗ G) ^λ |X ^τ ](y, ~ ξ)

=



 

 



 

 

 E ^anw

^λ



F  y + · 2

¹²



G  y − · 2

¹²

   

X τ



(~ ξ ), λ ∈ C + ;

E ^anf

^q



F  y + · 2

¹²



G  y − · 2

¹²

   

X _τ



(~ ξ), λ = −iq, q ∈ R − {0}

if they exist for s-a.e. y ∈ C 0 (B).

Using similar method in the proof Theorem 4.5 in [2], we have the following theorem.

Theorem 9. Let F ₁ ∈ A ^{(p n,s}

¹

⁾ and F ₂ ∈ A ^{(p n,s}

²

⁾ be given by (3) with replacing f by f ₁ and f ₂ , respectively, and let _p ¹

1

+ _p ¹

′

1

= 1(1 ≤ p 1 , p ₂ ≤

∞). Let X ^τ be given as in Lemma 3 and let t _p

∗

−1 < s ≤ t ^p

^∗

for some p ^∗ ∈ {1, . . . , k}. Then, for λ in C + and for s-a.e. ~ ξ ∈ B ^k , [(F ₁ ∗ F 2 ) λ |X ^τ ](y, ~ ξ) exists for s-a.e. y ∈ C 0 (B). Moreover, when t _p

∗

−1 < s < t _p

∗

, we have

[(F ₁ ∗ F 2 ) λ |X ^τ ](y, ~ ξ) = Z

R

ⁿ

f ₁

 w ~ y + ~ w _{~ ξ} + ~l 2

¹²

 f ₂

 w ~ y − ~ w _{ξ ~} − ~l 2

¹²



φ _λΓ (~l)d~l

with [(F 1 ∗F ² ) λ |X ^τ ](·, ~ξ) ∈ A ^{(1) n,s} if p ₂ ≤ p ^′ 1 and [(F 1 ∗F ² ) λ |X ^τ ](·, ~ξ) ∈ A ^{(p n,s}

²

⁾ if p ₂ ≥ p ^′ 1 , where Γ, φ _λΓ , w ~ _{~ ξ} and w ~ y are given by (4), (5), (6) and (7), respectively, and when s = t _p

∗

, we have

[(F ₁ ∗ F 2 ) _λ |X ^τ ](y, ~ ξ) =

 F ₁  1

2

¹²

(y + [~ ξ])



F ₂  1

2

¹²

(y − [~ξ])



. The following corollary follows from Theorem 9 immediately.

Corollary 10. Let F ₁ , F ₂ ∈ ∪ 1≤p≤∞ A ^{(p) n,s} . Let X τ be given as in Lemma 3 and let s = t p

^∗

for some p ^∗ ∈ {1, . . . , k}. Then, for λ in C ^∼ +

and for s-a.e. ~ ξ ∈ B ^k , [(F ₁ ∗ F 2 ) _λ |X τ ](y, ~ ξ ) exists for s-a.e. y ∈ C 0 (B) and it is given by

[(F ₁ ∗ F 2 ) _λ |X ^τ ](y, ~ ξ) =

 F ₁  1

2

¹²

(y + [~ ξ])



F ₂  1

2

¹²

(y − [~ξ])



. Theorem 11. Let X τ be given as in Lemma 3 and let t _p

∗

−1 < s < t _p

∗

for some p ^∗ ∈ {1, . . . , k}. For s-a.e. ~ξ in B ^k and for λ in C ^∼ ₊ , we have the followings:

1. if F ₁ , F ₂ ∈ A ^{(1) n,s} , then [(F ₁ ∗ F 2 ) _λ |X ^τ ](·, ~ξ) ∈ A ^{(1) n,s} , 2. if F ₁ , F ₂ ∈ A ^{(2) n,s} , then [(F ₁ ∗ F 2 ) λ |X ^τ ](·, ~ξ) ∈ A ^{(0) n,s} ,

3. if F ₁ ∈ A ^{(1) n,s} and F ₂ ∈ A ^{(2) n,s} , then [(F ₁ ∗ F 2 ) _λ |X τ ](·, ~ξ) ∈ A ^{(2) n,s} , 4. if F ₁ ∈ A ^{(1) n,s} and F ₂ ∈ A ^{(1) n,s} ∩ A ^{(2) n,s} , then [(F ₁ ∗ F 2 ) λ |X ^τ ](·, ~ξ) ∈

A ^{(1) n,s} ∩ A ^{(2) n,s} ,

5. if F ₁ ∈ A ^{(1) n,s} and F ₂ ∈ A ^{(0) n,s} , then [(F ₁ ∗ F 2 ) λ |X ^τ ](·, ~ξ) ∈ A ^{(0) n,s} . Proof. Let F 1 , F ₂ be given by (3) with replacing f by f 1 , f ₂ , respec- tively. For λ in C ₊ let

h(λ, ~u, ~ w _{ξ ~} ) = Z

R

ⁿ

f ₁

 ~ u + ~ w _{ξ ~} + ~l 2

¹²

 f ₂

 ~ u − ~ w _{ξ ~} − ~l 2

¹²



φ _λΓ (~l)d~l where ~u in R ⁿ and Γ, φ _λΓ , ~ w _{~ ξ} are given by (4), (5), (6), respectively.

1. By Theorem 9, it suffices to show that h(λ, ·, ~ w _{~ ξ} ) is in L ₁ (R ⁿ ) for λ ∈ C ^∼ + . But this fact follows from the following;

Z

R

ⁿ

|h(λ, ~u, ~ w _{ξ ~} )|d~u

≤  |λ|Γ 2π



ⁿ

2

Z

R

ⁿ

Z

R

ⁿ

   

f ₁

 ~ u + ~ w _{~ ξ} + ~l 2

¹²

 f ₂

 ~ u − ~ w _{~ ξ} − ~l 2

¹²

   

d~ld~ u

=  |λ|Γ 2π



ⁿ

2

Z

R

ⁿ

Z

R

ⁿ

   

f ₁



~v ₁ + w ~ _{ξ ~} 2

¹²

 f ₂



~v ₂ − w ~ _{~ ξ} 2

¹²

   

d~v ₁ d~v ₂

=  |λ|Γ 2π



ⁿ₂

kf 1 k 1 kf 2 k 1 , where ^{~ u+~l}

2

¹²

= ~v ₁ and ^{~ u−~l}

2

¹²

= ~v ₂ , by the change of variable theorem.

2. Note that, for λ ∈ C ^∼ + , h(λ, ·, ~ w _{ξ ~} ) is L _∞ (R ⁿ ) since

|h(λ, ~u, ~ w _{~ ξ} )| ≤  |λ|Γ 2π



ⁿ₂

Z

R

ⁿ

   

f ₁

 ~ u + ~ w _{~ ξ} + ~l 2

¹²

 f ₂

 ~ u − ~ w _{~ ξ} − ~l 2

¹²

   

d~l

≤  |λ|Γ π



ⁿ

2

kf 1 k 2 kf 2 k 2

for ~u ∈ R ⁿ by H¨older inequality. Thus we have h(λ, ·, ~ w _{~ ξ} ) ∈ C ⁰ (R ⁿ ) from a standard argument.

3. By Theorem 9, it suffices to show that h(λ, ·, ~ w _{~ ξ} ) is in L ₂ (R ⁿ ) for λ ∈ C ^∼ + . But this fact follows from the following;

Z

R

ⁿ

|h(λ, ~u, ~ w _{~ ξ} )| ² d~ u

≤  |λ|Γ 2π

 n Z

R

ⁿ

Z

R

ⁿ

   

f ₁

 ~ u + ~ w _{~ ξ} + ~l 2

¹²

 f ₂

 ~ u − ~ w _{ξ ~} − ~l 2

¹²

   

d~l

 2

d~ u

=  |λ|Γ 2π

 n Z

R

ⁿ

Z

R

ⁿ

   

f ₁

 ~ u + ~ w _{~ ξ} + ~l 2

¹²

 f ₂

 ~ u − ~ w _{ξ ~} − ~l 2

¹²

   

d~l



×

Z

R

ⁿ

   

f ₁

 ~ u + ~ w _{ξ ~} + ~z 2

¹²

 f ₂

 ~ u − ~ w _{ξ ~} − ~z 2

¹²

   

d~ z

 d~ u.

Let ~r = ^{~ u+~l}

2

¹²

and ~s = ^{~ u+~ z}

2

¹²

. Then we have Z

R

ⁿ

|h(λ, ~u, ~ w _{ξ ~} )| ² d~ u

≤  |λ|Γ 2π

 n

2 ⁿ Z

R

ⁿ

|f 1 (~r + 2 ⁻

¹²

w ~ _{~ ξ} )|

Z

R

ⁿ

|f 1 (~s + 2 ⁻

¹²

w ~ _{ξ ~} )|

× Z

R

ⁿ

|f ² ( √

2~u − ~r − 2 ⁻

¹²

w ~ _{~ ξ} )||f ² ( √

2~u − ~s − 2 ⁻

¹²

w ~ _{ξ ~} )|d~ud~sd~r

≤  |λ|Γ 2π

 n

2

ⁿ²

kf 1 k ² 1 kf 2 k ² 2

by H¨older inequality.

4. It follows from 1 and 3.

5. It follows immediately and it is trivial.

By Theorems 5 and 9, using similar method in the proof of Theorem 4.8 in [2], we have the following theorem.

Theorem 12. Let F ₁ , F ₂ ∈ ∪ 1≤p≤∞ A ^{(p) n,s} be given by (3) with replac- ing f by f ₁ , f ₂ , respectively, and let X _τ be given as in Lemma 3. Then, for λ in C ₊ and for s-a.e. (~ ξ ₁ , ~ ξ ₂ ) ∈ B ^k × B ^k , we have

T _λ [[(F ₁ ∗ F 2 ) _λ |X ^τ ](·, ~ξ 1 )|X ^τ ](y, ~ ξ ₂ )

=



T _λ [F 1 |X ^τ ]  y

2

¹²

, ξ ~ ₂ + ~ ξ ₁ 2

¹²



T _λ [F 2 |X ^τ ]  y

2

¹²

, ξ ~ ₂ − ~ξ ¹ 2

¹²



, for s-a.e. y ∈ C 0 (B).

The following theorem shows that the conditional Fourier-Feynman transform of conditional convolution product of some cylinder type func- tions is a product of conditional transforms. For convenience, if λ =

−iq(q ∈ R−{0}), then we denote [(F 1 ∗F 2 ) _λ |X ^τ ](·, ~ξ 1 ) by [(F ₁ ∗F 2 ) q |X ^τ ](·, ξ ~ ₁ ) in the following theorem.

Theorem 13. Let X τ be given as in Lemma 3 and let q be a non-zero real number.

1. Let F ₁ , F ₂ ∈ A ^{(1) n,s} . Then, for s-a.e. (~ ξ ₁ , ~ ξ ₂ ) ∈ B ^k × B ^k , we have T _q ⁽¹⁾ [[(F ₁ ∗ F 2 ) q |X ^τ ](·, ~ξ 1 )|X ^τ ](y, ~ ξ ₂ )

=



T _q ⁽¹⁾ [F ₁ |X ^τ ]  y 2

¹²

,

~ ξ ₂ + ~ ξ ₁ 2

¹²



T _q ⁽¹⁾ [F ₂ |X ^τ ]  y 2

¹²

,

ξ ~ ₂ − ~ξ 1

2

¹²



for s-a.e. y ∈ C 0 (B).

2. Let F ₁ ∈ A ^{(1) n,s} and F ₂ ∈ A ^{(2) n,s} . Then, for s-a.e. (~ ξ ₁ , ~ ξ ₂ ) ∈ B ^k × B ^k , we have

T _q ⁽²⁾ [[(F ₁ ∗ F 2 ) q |X ^τ ](·, ~ξ 1 )|X ^τ ](y, ~ ξ ₂ )

=



T _q ⁽¹⁾ [F ₁ |X ^τ ]  y 2

¹²

,

~ ξ ₂ + ~ ξ ₁ 2

¹²



T _q ⁽²⁾ [F ₂ |X ^τ ]  y 2

¹²

,

ξ ~ ₂ − ~ξ 1

2

¹²



for s-a.e. y ∈ C ⁰ (B).

3. Let F 1 ∈ A ^{(1) n,s} and F ₂ ∈ A ^{(1) n,s} ∩ A ^{(2) n,s} . Then, for s-a.e. (~ ξ ₁ , ~ ξ ₂ ) ∈ B ^k × B ^k , we have

T _q ⁽¹⁾ [[(F ₁ ∗ F 2 ) q |X ^τ ](·, ~ξ 1 )|X ^τ ](y, ~ ξ ₂ )

=



T _q ⁽¹⁾ [F ₁ |X ^τ ]  y

2

¹²

, ~ ξ ₂ + ~ ξ ₁ 2

¹²



T _q ⁽¹⁾ [F ₂ |X ^τ ]  y

2

¹²

, ξ ~ ₂ − ~ξ ¹ 2

¹²



and

T _q ⁽²⁾ [[(F 1 ∗ F ² ) q |X ^τ ](·, ~ξ ¹ )|X ^τ ](y, ~ ξ ₂ )

=



T _q ⁽¹⁾ [F ₁ |X τ ]  y

2

¹²

, ~ ξ ₂ + ~ ξ ₁ 2

¹²



T _q ⁽²⁾ [F ₂ |X τ ]  y

2

¹²

, ξ ~ ₂ − ~ξ 1

2

¹²



for s-a.e. y ∈ C 0 (B).

Proof. The results follow from Corollary 10, Theorems 6, 11 and 12.

5. Stability theories

Let H be an infinite dimensional separable real Hilbert space and let

∆ n = {(s 1 , s ₂ , . . . , s _n ) ∈ [0, T ] ⁿ : 0 = s ₀ < s ₁ < s ₂ < · · · < s ⁿ ≤ T } for any fixed n ∈ N.

Let M

^′′

n = M

^′′

n (∆ n × H ⁿ ) be the class of all complex Borel measures on ∆ n × H ⁿ and let kµk = var µ, the total variation of µ in M

^′′

n . Let S _n,B ^′′ = S _n,B ^′′ (∆ _n × H ⁿ ) be the space of functions of the form

F _n (x) = Z

∆

n

×H

ⁿ

exp

 i

n

X

j=1

(h j , x(s j )) ^∼



dµ _F

_n

(~s,~h) (11)

for s-a.e. x ∈ C 0 (B), where ~s = (s ₁ , . . . , s _n ), ~h = (h ₁ , . . . , h _n ) and µ F

n

∈ M

^′′

n . Here we take kF ⁿ k

^′′

n = inf{kµ ^F

n

k}, where the infimum is taken over all µ F

n

’s so that F n and µ F

n

are related by (11).

Let M ^′′ = M ^′′ (P ∆ n × H ⁿ ) be the class of all sequences {µ ⁿ } of measures such that each µ n ∈ M

^′′

n and P _∞

n=1 kµ ⁿ k < ∞. Let S _B ^′′ = S _B ^′′ (P ∆ n × H ⁿ ) be the space of functions of the form

F (x) =

∞

X

n=1

F _n (x),

(12)

s -a.e. x ∈ C 0 (B), where each F n ∈ S _n,B ^′′ and P _∞

n=1 kF ⁿ k

^′′

n < ∞. The norm of F is defined by kF k

^′′

= inf{ P _∞

n=1 kF ⁿ k

^′′

n }, where the infimum is taken over all representations of F given by (12).

Note that if n and l are positive integers then S _n,B ^′′ ⊂ S _n+l,B ^′′ and S _n,B ^′′ ⊂ S _B ^′′ for all n ∈ N. Moreover we can show that S _n,B ^′′ is a Banach space and S _B ^′′ is a Banach algebra.

Theorem 14. Let F n ∈ S _n,B ^′′ be given by (11) and let X τ be given as in Lemma 3. Let q ∈ R − {0} and 1 ≤ p ≤ ∞. Then, for s-a.e. ~ξ in B ^k , T _q ^(p) [F n |X ^τ ](y, ~ ξ ) exists for s-a.e. y ∈ C 0 (B) and it is given by

T _q ^(p) [F n |X ^τ ](y, ~ ξ) (13)

= X

j

1

+···+j

k

=n

Z

∆

^′n;j1,...,jk

×H

ⁿ

H _n (y, ~ ξ, ~s, ~h)G n (−iq, ~τ, ~s,~h)dµ ^F

n

(~s,~h)

where for j ₁ + · · · + j ^k = n

~s = (s _1,1 , . . . , s _1,j

₁

, s _2,1 , . . . , s _2,j

₂

, . . . , s _k,1 , . . . , s k,j

_k

);

(14)

∆ ^′ _n;j

₁

_,...,j

_k

(15)

= {~s : 0 = s ^1,0 < s _1,1 < · · · < s ^1,j

1

≤ t ¹ < s _2,1 < · · · < s ^2,j

2

≤ t 2 < · · · ≤ t k−1 < s _k,1 < · · · < s ^k,j

k

≤ t ^k = T },

(16) ~h = (h _1,1 , . . . , h _1,j

₁

, h _2,1 , . . . , h _2,j

₂

, . . . , h _k,1 , . . . , h k,j

_k

);

(17) H _n (y, ~ ξ, ~s, ~h) = exp

 i

k

X

α=1 j

α

X

β=1

(h α,β , y(s α,β ) + [~ ξ](s α,β )) ^∼

 ,

t ₀ = s _1,0 = 0, t α = s _α+1,0 = s α,j

α

+1 (α = 1, . . . , k − 1), (18)

s _k,j

_k

₊₁ = t k = T ;

(19) l _α,v = s α,v − s ^α,v−1 , for α = 1, . . . , k; for v = 1, . . . , j α + 1,

~

τ = (t 1 , . . . , t _k );

(20)

G n (λ, ~τ, ~s,~h) (21)

= exp



− 1 2λ

k

X

α=1 j

α

+1

X

v=1

l _α,v    

v−1

X

β=1

t _α−1 − s ^α,β t _α − t ^α−1 h _α,β

+

j

α

X

β=v

t α − s α,β

t _α − t α−1

h _α,β    

2 

with λ ∈ C ^∼ + .

Proof. For λ > 0, for s-a.e. ~ ξ in B ^k and for s-a.e. y in C ₀ (B), we have E[F n (y + λ ⁻

¹²

(x − [x]) + [~ξ])]

= Z

∆

n

×H

ⁿ

exp

 i

n

X

j=1

(h j , y(s j ) + [~ ξ](s j )) ^∼

 Z

C

₀

(B)

exp

 iλ ⁻

¹²

n

X

j=1

(h j ,

x(s j ) − [x](s ^j )) ^∼



dm _B (x)dµ F

n

(~s,~h)

by Fubini theorem where ~s = (s 1 , . . . , s _n ) and ~h = (h 1 , . . . , h _n ). Let ~s,

∆ ^′ _n;j

1

,...,j

k

, ~h and H n be given by (14), (15), (16) and (17), respectively.

Then, by Lemma 2, we have

E[F n (y + λ ⁻

¹²

(x − [x]) + [~ξ])]

= X

j

1

+···+j

k

=n

Z

∆

^′n;j1,...,jk

×H

ⁿ

H _n (y, ~ ξ, ~s, ~h)

k

Y

α=1

Z

B

^jα+1

exp

 iλ ⁻

¹²

j

α

X

β=1

 h _α,β ,

β

X

v=1

pl α,v x α,v − s _α,β − t α−1

t _α − t α−1 j

α

+1

X

v=1

pl α,v x α,v

 ∼ 

dm ^j

^α

⁺¹ (~x _α )



dµ _F

_n

(~s,~h)

by Wiener integration theorem (Theorem 1) where ~x α = (x _α,1 , . . . , x α,j

α

+1 ) and l α,v is given by (19) with (18). Let ~τ and G n be given by (20) and (21), respectively. Then we have

E[F n (y + λ ⁻

¹²

(x − [x]) + [~ξ])]

= X

j

₁

+···+j

k

=n

Z

∆

^′n;j1,...,jk

×H

ⁿ

H n (y, ~ ξ, ~s, ~h)

k

Y

α=1

Z

B

^jα+1

exp

 iλ ⁻

¹²

j

α

+1

X

v=1

 pl _α,v

 v−1

X

β=1

t _α−1 − s ^α,β t α − t α−1

h _α,β +

j

α

X

β=v

t _α − s ^α,β t α − t α−1

h _α,β

 , x _α,v

 ∼ 

dm ^j

^α

⁺¹ (~x α )



dµ F

n

(~s,~h)

= X

j

₁

+···+j

k

=n

Z

∆

^′n;j1,...,jk

×H

ⁿ

H n (y, ~ ξ, ~s, ~h)G n (λ, ~τ, ~s,~h)dµ F

n

(~s,~h) since (h, ·) ^∼ is normally distributed with mean 0 and variance |h| ² (h 6=

0). By Morera’s theorem, we have T _λ [F n |X ^τ ](y, ~ ξ)

= X

j

1

+···+j

k

=n

Z

∆

^′n;j1,...,jk

×H

ⁿ

H n (y, ~ ξ, ~s, ~h)G n (λ, ~τ, ~s,~h)dµ F

n

(~s,~h) for λ ∈ C ⁺ . For 1 ≤ p ≤ ∞ let T ^{q (p)} [F n |X ^τ ](y, ~ ξ) be given by (13). When p = 1 we have

|T ^λ [F n |X ^τ ](y, ~ ξ ) − T q ⁽¹⁾ [F n |X ^τ ](y, ~ ξ )|

≤ X

j

1

+···+j

k

=n

Z

∆

^′n;j1,...,jk

×H

ⁿ

|G ⁿ (λ, ~τ, ~s,~h) − G ⁿ (−iq, ~τ, ~s,~h)|

d kµ ^F

n

k(~s,~h)

and when 1 < p ≤ ∞, for ¹ p + _p ¹

_′

= 1 and γ > 0, we have Z

C

0

(B) |T ^λ [F n |X ^τ ](γy, ~ ξ ) − T q ^(p) [F n |X ^τ ](γy, ~ ξ )| ^p

^′

dm _B (y)

≤

 X

j

1

+···+j

k

=n

Z

∆

^′n;j1,...,jk

×H

ⁿ

|G ⁿ (λ, ~τ, ~s,~h) − G ⁿ (−iq, ~τ, ~s,~h)|

d kµ ^F

n

k(~s,~h)

 p

^′

.

Letting λ → −iq through C + , we have the results by the dominated convergence theorem.

Theorem 15. Let F ∈ S _B ^′′ be given by (12) and let X τ be given as in Lemma 3. Let q ∈ R − {0} and 1 ≤ p ≤ ∞. Then, for s-a.e. ~ξ in B ^k , T q ^(p) [F |X ^τ ](y, ~ ξ ) exists for s-a.e. y ∈ C ⁰ (B) and it is given by

T _q ^(p) [F |X ^τ ](y, ~ ξ) =

∞

X

n=1

T _q ^(p) [F n |X ^τ ](y, ~ ξ)

(22)

where T q ^(p) [F n |X ^τ ] is given by (13) in Theorem 14.

Proof. Without loss of generality, we can assume P ∞

n=1 kµ ^F

n

k < ∞ where F n and µ F

n

are related by (11). For λ > 0, for s-a.e. ~ ξ ∈ B ^k and for s-a.e. y ∈ C 0 (B), we have

T _λ [F |X ^τ ](y, ~ ξ) =

∞

X

n=1

T _λ [F n |X ^τ ](y, ~ ξ) (23)

by Theorem 14 and the dominated convergence theorem. Since P ∞

n=1 T _λ [F n |X ^τ ](y, ~ ξ) converges uniformly on C + by Theorem 14, we have (23) for all λ ∈ C ⁺ . For 1 ≤ p ≤ ∞ let T ^{q (p)} [F |X ^τ ](y, ~ ξ) be given by (22). When p = 1 we have

|T λ [F |X τ ](y, ~ ξ ) − T q ⁽¹⁾ [F |X τ ](y, ~ ξ )|

≤

∞

X

n=1

X

j

1

+···+j

k

=n

Z

∆

^′n;j1,...,jk

×H

ⁿ

|G n (λ, ~τ, ~s,~h) − G n (−iq, ~τ, ~s,~h)|

d kµ F

n

k(~s,~h)

and when 1 < p ≤ ∞, for ¹ p + _p ¹

_′

= 1 and γ > 0, we have Z

C

0

(B) |T ^λ [F |X ^τ ](γy, ~ ξ ) − T q ^(p) [F |X ^τ ](γy, ~ ξ )| ^p

^′

dm _B (y)

≤

 ∞

X

n=1

X

j

₁

+···+j

k

=n

Z

∆

^′n;j1,...,jk

×H

ⁿ

|G ⁿ (λ, ~τ, ~s,~h) − G ⁿ (−iq, ~τ, ~s,~h)|

d kµ ^F

n

k(~s,~h)

 p

^′

.

Letting λ → −iq through C ⁺ , we have the results by the dominated convergence theorem.

Let M(H) be the class of all complex Borel measures on H and let G be the set of all C-valued functions θ on [0, T ] × B which have the following form

θ(s, y) = Z

H exp{i(h, y) ^∼ }dσ ^s (h) (24)

where {σ ^s : s ∈ [0, T ]} is the family from M(H) satisfying the following conditions;

1. for each Borel subset E of H, σ ^s (E) is a Borel measurable function

of s on [0, T ],

2. kσ ^s k ∈ L 1 ([0, T ]).

Let θ ∈ G be given by (24) and for s-a.e. x in C 0 (B) let F n (x) =

Z T 0

θ(s, x(s))ds

 ⁿ (25)

and

F (x) = exp

Z T 0

θ(s, x(s))ds

 (26)

where n is any fixed natural number. For any Borel measurable subset E of (0, T ] × H, let

µ(E) = Z T

0

σ _s (E ^(s) )ds,

where E ^(s) = {h ∈ H : (s, h) ∈ E}. By unsymmetric Fubini theorem ([8]), we have, for s-a.e. x in C 0 (B),

F ₁ (x) = Z T

0

θ(s, x(s))ds = Z T

0

Z

H exp{i(h, x(s)) ^∼ }dσ ^s (h)ds

= Z

(0,T ]×H exp{i(h, x(s)) ^∼ }dµ(s, h), and kµk ≤ R T

0 kσ ^s kds, so that F 1 ∈ S _1,B ^′′ ⊂ S _B ^′′ . Moreover, we can show that F n ∈ S _n,B ^′′ for each n ∈ N. Thus we have the following theorems by Theorems 14 and 15.

Theorem 16. Let F n be given by (25) and let X τ be given as in Lemma 3. Let q ∈ R − {0} and 1 ≤ p ≤ ∞. For any Borel subset E of

∆ n × H ⁿ let

µ ^′ _F

_n

(E) = n!

Z

∆

n

Z

H

ⁿ

χ _E (~s,~h)d

 n

Y

j=1

σ _s

_j

 (~h)d~s

where ~s = (s ₁ , . . . , s n ). Then, for s-a.e. ~ ξ in B ^k , T q ^(p) [F n |X ^τ ](y, ~ ξ) exists for s-a.e. y ∈ C ⁰ (B) and it is given by (13) with replacing µ F

n

by µ ^′ _F

n

. Theorem 17. Let F be given by (26) and let X τ be given as in Lemma 3. Let q ∈ R − {0} and 1 ≤ p ≤ ∞. Then, for s-a.e. ~ξ in B ^k , T _q ^(p) [F |X ^τ ](y, ~ ξ ) exists for s-a.e. y ∈ C 0 (B) and it is given by

T _q ^(p) [F |X ^τ ](y, ~ ξ) = 1 +

∞

X

n=1

1

n! T _q ^(p) [F n |X ^τ ](y, ~ ξ)

where T _q ^(p) [F n |X ^τ ] is given as in Theorem 16.

Let F(B) be the class of all functions of the form φ(x ₁ ) =

Z

H exp{i(h, x 1 ) ^∼ }dν(h) (27)

for s-a.e. x 1 in B where ν ∈ M(H). For s-a.e. x in C ⁰ (B) let K _n (x) = F n (x)φ(x(T )) and K(x) = F (x)φ(x(T )) (28)

where F n and F are given by (25) and (26), respectively. Then for λ > 0, x, y ∈ C 0 (B) and ~ ξ = (ξ ₁ , . . . , ξ _k ) ∈ B ^k , we have

|φ(y(T ) + λ ⁻

¹²

(x(T ) − [x](T )) + [~ξ](T ))|

(29)

= |φ(y(T ) + ξ k )| ≤ kνk.

Thus we have the following theorem.

Theorem 18. Let K n , K be given by (28) and let X τ be given as in Lemma 3. Let q ∈ R − {0} and 1 ≤ p ≤ ∞. Then, for s-a.e. ~ξ in B ^k , both T q ^(p) [K n |X ^τ ](y, ~ ξ) and T q ^(p) [K|X ^τ ](y, ~ ξ ) exist for s-a.e. y ∈ C 0 (B), and they are given by

T _q ^(p) [K n |X ^τ ](y, ~ ξ) = T _q ^(p) [F n |X ^τ ](y, ~ ξ)φ(y(T ) + ξ k ) and

T _q ^(p) [K|X ^τ ](y, ~ ξ) = T _q ^(p) [F |X ^τ ](y, ~ ξ)φ(y(T ) + ξ k )

= φ(y(T ) + ξ k ) +

∞

X

n=1

1

n! T _q ^(p) [K n |X ^τ ](y, ~ ξ), where T q ^(p) [F n |X ^τ ] and T q ^(p) [F |X ^τ ] are given as in Theorems 16 and 17, respectively.

Let ζ be a complex-valued Borel measure on [0, T ]. Then ζ = µ + ν d

can be decomposed uniquely into the sum of a continuous measure µ and a discrete measure ν d ([6, p.142]). Let δ τ

j

denote the Dirac measure with total mass one concentrated at τ j .

Let G ^∗ be the set of all C-valued functions θ on [0, T ] × B which have the form (24) where {σ ^s : s ∈ [0, T ]} is the family from M(H) satisfying the following conditions;

1. for each Borel subset E of H, σ ^s (E) is a Borel measurable function of s on [0, T ],

2. kσ ^s k ∈ L ¹ ([0, T ], B([0, T ]), kζk).

In convenience, let

ζ = µ +

r

X

j=1

w _j δ _τ

_j

(30)

where 0 ≤ τ 1 < · · · < τ ^r ≤ T and the w ^j ’s are in C for j = 1, . . . , r(∈ N), and let θ ∈ G ^∗ be given by (24). For s-a.e. x in C 0 (B) let

F _n (x) =

Z T 0

θ(s, x(s))dζ(s)

 n

(31) and

F (x) = exp

Z T 0

θ(s, x(s))dζ(s)

 (32) .

Theorem 19. Let F n be given by (31) and let X τ be given as in Lemma 3. Let q ∈ R − {0} and 1 ≤ p ≤ ∞. By reordering τ ^j ’s, t _j ’s and renaming τ j ’s by τ α,u ’s (α = 1, . . . , k; u = 1, . . . , r α ) let 0 ≤ τ 1,1 <

τ _1,2 < · · · < τ ^1,r

1

≤ t ¹ < τ _2,1 < · · · < τ ^2,r

2

≤ t ² < · · · ≤ t k−1 < τ _k,1 <

· · · < τ ^k,r

k

≤ t ^k = T , where r ₁ + · · · + r ^k = r. Then, for s-a.e. ~ ξ in B ^k , T q ^(p) [F n |X ^τ ](y, ~ ξ ) exists for s-a.e. y ∈ C 0 (B) and it is given by

T _q ^(p) [F n |X ^τ ](y, ~ ξ) (33)

= n! X

q

₁

+···+q

k

=n k

Y

α=1



X

l

₀

+l

1

+···+l

_rα

=q

α

L(q α )W (q α )



X

j

₁

+···+j

_rα+1

=l

0

Z

∆

l0;j1,...,jrα+1

Z

H

^qα

H q

α

(y, ~s,~h,~h ^′ , ~ ξ, ~ τ α )G q

α

(−iq, ~s,~h,~h ^′ , ~ τ α )

d(σ ~ s × σ ^{~ τ}

α

)(~h, ~h ^′ )dµ ^l

⁰

(~s)



where for q ₁ + · · ·+q k = n and for α = 1, . . . , k; for l ₀ + l ₁ + · · ·+l r

α

= q _α L(q α ) = 1

l ₁ ! · · · l ^r

α

! (34)

and

w _α,u = w _r

₁

_+···+r

_α−1

_+u (u = 1, . . . , r _α );

(35)

W (q α ) =

r

α

Y

u=1

w ^l _α,u

^u

;

(36)

for j ₁ + · · · + j ^r

α

+1 = l ₀

~s = (s _0,1 , . . . , s _0,j

₁

, s _1,1 , . . . , s _1,j

₂

, . . . , s _r

_α

_,1 , . . . , s _r

_α

_,j

_rα+1

);

(37)

∆ l

₀

;j

1

,...,j

_rα+1

= {~s : 0 ≤ t α−1 ≤ s 0,1 ≤ · · · ≤ s 0,j

1

≤ τ α,1 < s _1,1 <

(38)

· · · < s 1,j

2

< τ _α,2 < · · · < τ α,r

α

≤ s r

α

,1 ≤ · · · ≤ s r

α

,j

_rα+1

≤ t α ≤ T },

~h = (h 0,1 , . . . , h _0,j

₁

, h _1,1 , . . . , h _1,j

₂

, . . . , h _r

_α

_,1 , . . . , h _r

_α

_,j

_rα+1

) (39)

and

~

τ _α = (τ α,1 , . . . , τ _α,r

_α

);

(40)

σ _{~ s} =

r

α

Y

u=0 j

u+1

Y

v=1

σ _s

_u,v

, σ _{~ τ}

_α

=

r

α

Y

u=1 l

u

Y

1

σ _τ

_α,u

(41)

and

~h ^′ = (h ^′ _1,1 , . . . , h ^′ _1,l

1

, h ^′ _2,1 , . . . , h ^′ _2,l

2

, . . . , h ^′ _r

_α

_,1 , . . . , h ^′ _r

α

,l

_rα

);

(42)

for u = 0, . . . , r α − 1

h _u,j

_u+1

₊₁ =

l

u+1

X

β=1

h ^′ _u+1,β , s _u,j

_u+1

₊₁ = τ _α,u+1 = s _u+1,0 , (43)

h _r

_α

_,j

_rα+1

₊₁ = 0 and s _r

_α

_,j

_rα+1

₊₁ = t _α ;

H _q

_α

(y, ~s,~h,~h ^′ , ~ ξ, ~ τ _α ) (44)

= exp

 i

r

α

X

u=0 j

u+1

+1

X

v=1

(h u,v , y(s u,v ) + [~ ξ](s u,v )) ^∼



and for a = 0, 1, . . . , r α ; for b = 1, . . . , j _a+1 + 1

β _a,b = s a,b − s a,b−1 with s _0,0 = t α−1 ;

(45)