Selected papers of Takeyuki Hida

is given by


where £t is the innovation of the X(t). Given such an expression, the probabilistic structure of {X(t)} is fully described by the pair {
B(t) input

unknown device

9(B,t) output.

5 We wish to determine the dynamical structure of the given unknown box; this means that we must analyze the functional
u = u(x, t),

t > 0, x e D domain in Rn.

This equation has been discussed by A.V. Balakrishnan (see [21]). Another typical example arises in population biology and has been developed by D. Dawson. A simple case can be described, in his terminology (see [22]) by the equation

m=Au

+ B

^

In these cases the solution should be a functional of B, so that we are given an interesting class of Brownian functionals. In addition we have to deal with multiplication by B(t) which is a generalized function. 4) Feynman's path integral. A wave function tp(t,x) in quantum mechanics may be expressed by the so-called path-integral of the form

1>{t,x)= I ei&L(x'i'')d'Vx,

t>0,

where L is the Lagrangian and where the integral in Vx extends over all possible paths including the classical path. In order to concretize the Vx, let us assume that Vx is a measure on the function space, each member of which is viewed, with this measure, as a sample path of a Gaussian Markov process. (Such an assumption seems to be reasonable from some viewpoints.) With this assumption the action f0 L(x, x, s)ds now involves an integral of x(s)2 which is quite awkward like B(s)2. Thus we are inspired to define an integral of B(s)2, and further led to deal with a class of generalized functionals of Brownian motion. These examples have been the motivation of our work, and at the same time they are the problems to which we wish to apply our results. There are two basic tools of the analysis in our work. One is the integral representation of L2-functionals of white noise. The idea has been described in the book [1] by the present author. Another important tool is the variation of entire, homogeneous functionals developed by P. Levy [3]. By using his idea we are able to introduce a class of generalized Brownian functionals and to analyze them. The main topics are shown in the following flow diagram.

6 FLOW DIAGRAM

White Noise Finite Dimensional Approximation

{6(f)}

L2(E-,iJi)

<-

Harmonic Analysis

OH

RKHS & = &(E, C) <-

Functional Analysis

l4n\L2(R")

Applications

Generalized Functional

Everything can be complexified.

7 §1. W h i t e noise We shall first discuss the probability distribution of {B(t);t a triple

G R1}. Start with

EcL2{Rl)dE*,

(1)

where E is a nuclear space dense in L?{R}) and where E* is the dual space for E. We usually take the basic nuclear space E to be the Schwartz space S, sometimes Do = U G C ° ° ; £ ( ± ) i G C°°} or VK = {£ G C°°; support of £ C K}. The canonical bilinear form connecting E and JS* will be denoted by (a;, £), x G £*, £ G E. A sample path of B(t) is a member of E*\ that is (B,£) is well defined where {B, £) should be understood as

- J B(t)£'(t)dt. It is a Gaussian random variable. Further we see that the system {(B, £); £ G E} is Gaussian and E{(B,£))=0, £ ( ( 5 , 0(B,v))=Ey = ff

for every e G £ , B(t)Z'(t)dt J

B(sW(s)ds]j

(tAs)Z'(t)r]'(s)dtds

= Jmv(t)dt,

Ln&E.

We observe the characteristic functional C(£), £ G E, of {5(i); t 6 i ? } given by C(0=-B{exp(i(B,0]}. It is a characteristic function evaluated at 1 of a Gaussian random variable with mean 0 and variance ||£|| 2 , || • || being the L 2 (i?)-norm. Hence (2)

C(0 = exp

-\u\r

i&E.

We can now appeal to the Bochner-Minlos theorem. Let B be the cr-field generated by the cylinder subsets of E* of the form {x; ((x, £ i ) , . . . , (a;, £„)) G B, B is a Borel subset of Rn, & , . . . , £ „ t E}, n = 1,2,.... Theorem 1. (Bochner-Minlos) Let C(£) be a functional on E which is i) continuous, ii) positive definite, and hi) C(0) = 1.

8 Then there exists a probability measure fi on the measurable space (E*, B) such that C(0= /

exV[i(x,0]dfi(x).

JE*

In addition, such a measure is unique. Definition. The measure space (E*, B, fi) determined by the characteristic functional (2) is called a white noise (W.N.). The W.N. is nothing but the probability distribution of {B(t);t are many variations of W.N. We indicate a couple of them.

G R}. There

1) The measure associated with (by Theorem 1) the characteristic functional

C(0 = exp

< ^ , , 22 2U\\

+ i(m,0

<J

> 0, me

E*,

is called the measure W.N. with mean m and variance a2. In case m = 0, such a measure is denoted by [ia. 2) Let the basic nuclear space E be taken to be V{n) — {C°°-functions on a unit circle S\}. We then have a triple, instead of (1), V{TT) C L 2 ( 5 J ) C V(TT)*

and Theorem 1 can still be applied to obtain a measure of the periodic W.N. Some properties of white noise Proposition 1.1. The collection {(x,£), £ G E} forms a Gaussian system with mean 0 and covariance T(£,rj) = (£, ??), where (•, •) stands for the inner product in L\R). Proposition 1.2. W.N. has independent value at every moment in the sense that the random variables (a:,£i) and {x,^) on (E*,B,fi) are independent whenever £i(£)£2(£) = 0. Moreover, (x,£i) and (x,^) are independent if and only if

(6,6) = o. Proofs of these propositions are straightforward. Proposition 1.3. Two measures / i a i and fia2 are singular if and only if o~\ ^ o
A x

-{ ^^00h^l{x^)2

=a

"}

9 The strong law of large numbers tells us that

while na2 (A) — 0. Thus the assertion follows. §2. Functionals of Brownian motion Since W.N. (E*,B,fi) is the probability distribution of {B[t)}, each member x of E* with measure fx can be viewed as a sample path of B(t). Hence a general Brownian functional may be expressed as
*)ei2(£',c). For notational convenience, we denote the Hilbert space L2(E*,fi) (L2).

simply as

First we introduce two classes of basic functionals of (L2). 1) Polynomials in x (e E*) Let P(t\,..., tn) be a polynomial in t = ( i i , . . . , tn) e Rn with complex coefficients, and let £ i , . . . ,£„ be linearly independent vectors in E. A functional |(^0|

2n

^(a:) = (2n-l)!!||^||2"
which proves that any polynomial in x belongs to (L2). 2) Exponential functionals A functional expressed in the form (4)


a £ C,

is called an exponential functional. Any exponential functional belongs to (L2). This is guaranteed by the following computation. J \ex.V[a{x,£)}\2d^{x)

=

J 'exp 2at-

t2

m\r<

dt,

Let A be an algebra generated by the exponential functionals: A = alg.{exp[a<x,£)];a € C, £ € E}.

a = a + ib.

10 Theorem 2. A is dense in (L2). Proof. To prove the theorem it suffices to establish the fact that (*) if f G (I/ 2 ) is orthogonal to every exponential functional of the form |Texp[itfc(a;,^fc)]

(finite product),

k

where £& G E and tk S R, then f — 0 a.e. Assume that / ,

Y[exp[itk(x,^k)]f{x)dfi(x)

= 0.

fe=i

This can be expressed as

/ ,

Y[exp[itk(x,£k)]E(
= 0,

where Bn is the cr-field generated by the (x,£k), 1 < k < n, since the conditional expectation E(-\Bn) gives an orthogonal projection down to L2(E*, 6„,/x). Now let the tfc's run through the real numbers. Then the theory of Fourier transform on L2(Rn) shows that EQp\Bn) = 0, a.e. Letting n —>• oo in such a way that {£ n } forms a base of L2(Rn), fix) = 0,

we have

a.e.

We then come to the algebra V generated by polynomials in x: V = alg. {polynomials}. Corollary 1. V is dense in (L2). Since A is dense in (L2) and since exp[a(a;, £)] is approximated by the polynomials, the assertion follows immediately. Indeed, exp[a(x,£)] has a Taylor expansion and the power series converges in (L2). Corollary 2. Let {£„} be a complete orthonormal system in L2(R). The subalgebra ?>({£«}) = ate-{IlLi[<*.&}]"*; nk > 0,3 = 1,2,...} o/ P ^ a^o dense in (L2). Since any £ in E is expressed as ( = ^ <

n

in L 2 (i?),

n

we have (x,0 = Y^an(x^n)

in (L 2 ).

11 With this one can show that any polynomial in x is approximated by polynomials based on (x,£n)'s; for instance

N-yoo

§3. Multiple Wiener integral (homogeneous chaos) Introduce a transformation T denned by (5)

(7»(0 = /

exp[i(x,Z)Mx)dfi(x),


JE*

The transformation T looks like a generalization of the ordinary Fourier transform, however it is somewhat different in its role. A crucial difference is that T carries (L2)-functionals not onto itself but to a functional space over E. Set

.F^T{(£ 2 )}EE{(T^)(0;¥>e(£ 2 )}. Then one can easily show that i) J- is a vector space, ii) T is an isomorphism of vector spaces (L2) and J-. Indeed, the second property comes from Theorem 2. We now wish to introduce an "inner product" so that T becomes a Hilbert space isomorphic to (L 2 ) under the transformation T. For this purpose T must be topologized so that J 7 is a reproducing kernel Hilbert space (R.K.H.S.) with reproducing kernel C(£ — rj). Before proving this fact, it seems to be better to pause here in order to have a quick review of the general theory of R.K.H.S. Let T be any set, and let K(t, s) be a positive definite function on TxT. Then there is a Hilbert space J-(K) consisting of functions on T such that a) for any fixed s, K(t, s) belongs to F(K) as a function of t, b) for any /(•) in F{K), (6)

(f(.),K(;s))

=

f(s),

where the ordinary bracket denotes the inner product introduced in

F(K).

c) F{K) is generated by the family {K(-, s); s £ T}. Steps for a construction of such a R.K.H.S. J-(K), when K(t,s) be described as follows:

is given, may

i) form a vector space T\ involving all the linear combinations of the form Y^,akK(-,Sk), afc: constants, ii) introduce a seminorm || Yl,akK(-,Sk)\\2 sense since K is positive definite),

= J2akajK(sk,Sj)

in T\ (this makes

12 iii) form the factor space J-\/X = Ti where X is the ideal consisting of the vectors with zero seminorm, and finally iv) let Ti be extended so as to be complete. We are now ready to discuss the isomorphism in question. Observe that T acts in such a way that exp[i(x, 77)] ->• / exp[i(x, £)] • exp[-i(a:, r))]d{j,(x) = C(£ - 77), ^a fc exp[i(a;,77fc)] ->• y^akC((,

+ rjk),

and evaluate the inner product in (L2):

/ '

exp[i(x,rn)} • exp[i(a;,772)] = C(rn - 772),

then we are led to define C

Q T akC(- + r,k), C{- -V))=Y.

^

+ »?*)•

Since the algebra A, generated by the exp[a(a;,77)], 77 G E, is dense, we must have

(/(•), C(.-rj))

= f{r,)

for general / in J- (see (6)). In other words, T is defined to be the R.K.H.S. with kernel C(£ — 77). Thus we have proved. Theorem 3. Suppose T is the R.K.H.S. with kernel C(£ — 77), £, 77 € E. Then T gives an isomorphism Now to fix the idea set E = S (the Schwartz space). Consider the formula (7)

+ \t2\\vf

•luf-^v)t

T(exp[i(x,77)i])(0=exp

and evaluate ^ | t _ 0 to get

T((x,V))(0=i(v,OC(0,

V£L2(R).

It is noted that (x, £) defines a random variable not only for ( e S, but also when £ is a member of L2(R). However, (x,£) with £ in L2(R) is not a continuous linear functional on S* but it is still a Gaussian random variable with mean 0 and variance ||£|| 2 . Because of this observation as well as (7) one has, via T,

{x,Z)^i(T),Z)C(0, N

N

l[(x,Vn) ^ iN Y[(r,nA)C(Z), i=l

n=l

2

VeL

(R)

13 where {qn} is an orthonormal system in L2(R). In general a functional of the form

•7

JF(h,..

.,tN)t(h)

• ••^(tN)dt1 • • • dtNC(0

with a symmetric L2 (RN )-iunction F can be found in the space J-. After these observations we come to the following decomposition of T. The reproducing kernel C(£ — rj) is expanded in the form (8)

C(t-T,)

=

Y,Cn(t,r,), n

where Cn(£,T,) =

C(^(C,v)nC(r,).

Obviously C n (£, 77) is positive definite, so that it defines a R.K.H.S., denoted by J-n, which is a subspace of T. In addition, we can prove that if n ^ m, then (Cn(-,r)),Cm(;r)')) this means that (9)

= 0,

for any r?, r/ € S,

Tn ± Tm,

n^m.

Summing up, we have Theorem 4. The R.K.H.S. J- has direct sum decomposition oo

(10)

F = Y, ®^. n=0

Remark. The orthogonal projection operator Pn to J-n is given by (11)

P n / ( 0 = (/(•), C n ( - , 0 ) ,

f o r / S ^ .

Since J is an isomorphism, we have subspaces of (L2) in such a way that '

\J n) — rln-

Theorem 4 becomes a direct sum decomposition of (L2): oo

(12)

(L2) = Y J ©% n

(Wiener-Ito decomposition).

n=0

Definition. The subspace Hn is called the Multiple Wiener integral of degree n. N o t e . Sometimes Hn is called the homogeneous chaos of degree n.

14 Alternative approach to the multiple Wiener integral. Let us recall a member of the algebra P({£ n }) generated by polynomials in x based on a fixed complete orthonormal system {£n}- It is dense in (L2) (see Corollary 2 in §2). Definition. A polynomial in P({£n}) expressed in the form const H Hkj f^lM.\

:

finite product

is called a Fourier-Hermite (F-H) polynomial (based on {£«}), where i? n 's are ordinary Hermite polynomials in single variable of degree n. The following assertions are straightforward from the results in §2 and from the basic properties of Hermite polynomials. Proposition 3.1. i) The space spanned by the F-H polynomials is dense in (L2). ii) F-H polynomials Y[ Hkj ( unless kj = lj for all j .

)i ) and FJ Hij (

S1 J are orthogonal in (L2)

iii) {c{fc.} J | • Hkj ( 7$ )} forms a complete orthonormal system in (L2), where C{k\ is a constant and its actual value is given by

Cft}

_

1

" x/TTW

Let 1-ln be the subspace of (L2) spanned by F-H polynomials of degree n. Proposition 3.1 enables us to obtain the following direct sum decomposition oo

2

(12')

(L ) = J2®Hn n=0

(see Cameron-Martin [6]). Remark. %n is independent of the choice of a complete orthonormal system {£„}. This fact is proved by the addition formula of the Hermite polynomials. Now to show (13)

__ Hn = Hn

(defined before)

we choose

(14)

^) = n^(^n)> $> = »•

15 From the definition of T we have (T
( ^ )

<*M*0

= yexp^x^OjII^PK^'^)]]!^ ( ^ )

^

£ = V ^ a ? ^ + £' (from the basis)

= | e x p ^ x , O I W I I / exp[*«ifo fc>] I I ^

(~f^)

^(a;)

(using independence)

= exp

-jiia'n^s^p

•^II^-II 2

(this equality will be established later) : exp

-W (v^os^n^'O'

where -F(i 1 ,..., tn) is the symmetrization of ^j1® ® £22® ® • • •. We claim that (15)

(Tip)® e Tn.

In fact, starting with Cn(-,rj) = C{-)C(jiy ,ri\ , r\ £ S, we have more general expression in the case rj = X^fc^fc- Then, if the limit is also taken into account, we immediately prove (15). To establish the formula (16)

/exp[ia(x, V )]H k

(^S)

dfi(x) = (V2i)kak exp

- y

Nl = i,

we recall the generating function of Hermite polynomials exp[2te — t2]. Set
ln.,,2

y/tofarjfidpix)

exp[iV2t^,r})}.

16 Since f(x) is expressed as

«'>-££*» we have established

which implies (16). Noting that .TVs are disjoint, we have proved (13). Let us now observe the L2(Rn)-norm of F which appeared in the above computation. Elementary calculation shows that

im = vTT*

2«/2

Vnf'

Since we have already proved that the (L 2 )-norm of (p given by (14) is \/X\kj we have the following result.

2" / ' 2 ,

Theorem 5. i) There is a one-to-one correspondence between tp £ "Hn and F £ (symmetric L2(Rn)-functions) determined by

L2(Rn)

(TV)(0 = inC{i) J... J F(h,..., *„)$(*!) • • • atn)dU • • • dtn, ii) IMI(L*) = VrH\\F\\LnRn). Definition. The expression in i) of Theorem 5 is said to be the integral representation of
(17)

(L2) * Y, ©Vni^CR").

which is identical with the so-called Fock space.

§4. Applications We shall illustrate, in this section and the next, how our results can be applied to various problems in probability theory. Much attention will be paid to the space V.2 which can be viewed as a class of quadratic functionals of Brownian motion. Specifically for the space %2 our method seems to be a powerful tool, and further

17 our observation provides a good interpretation of why we are led to introduce the class of generalized Brownian functionals which will be discussed in §9. Let ip(x) be a real-valued %2-functional. Then the kernel F of its integral representation is a real-valued L2(.R2)-function. It defines an integral operator, and one can therefore appeal to the Hilbert-Schmidt expansion theorem to obtain (18)

F(u, v) = J2 T-Vni^Vniv) An

L2(R2),

in

where {A„, 7?n; n > 1} is the eigensystem of F: (19)

A„ / F{u,v)r/n(v)dv

= r)n(u),

n > 1,

with ||?7n|| = 1. Associated with the expansion (18) is an expansion of
^ ) = E^^r?»)2-1}-

(20)

in L2

( )>

where the right-hand side is a sum of independent random variables. Theorem 6. Any real-valued random variable ip in Tf.2 is expressed in the form (20) and its characteristic function ^ ( z ) , z € R, is given by (21)

x*{z) =

5{2iz;F)-l'\

where S(2iz; F) is the modified Fredholm determinant of F. Note. The modified Fredholm determinant is defined to be 0 F(w 2 ,wi) n=0

J

F(ui,u2) 0

••• •••

F(ui,un) F(u2,un)

dun.

J

F{un,ui)

•••

•••

0

Proof of Theorem 6. The expansion for (

• ) - / exp

"E^" 1 ^-^ 2 - 1 )

J|<^exp[-izA n 1 ] /

dfx(x)

2TT

-,1/2 1

1

n{(l-2^A- )exp[-2iZA- ]} 6{2iz;F)-^2.

21

1

exp[iz

exp

dx

18 Corollary 1. Let

of the random

7i=0, n

Corollary 2. Assume that the F corresponding to the ip in Theorem 6 further satisfies (22)

F(-u,-v) = -F(u,v).

Then a) if A is an eigenvalue, so is —A, b) the probability distribution of ip is symmetric and - ' • ! ' •

U„>0 c) 72 P +i = 0,

l2p

= 2^(2P-iy.J2 K2p, P > I A„>o

Example 1. N. Wiener's model of the "brain wave" is described as in 2) of §0, where an %2-functional is used to express the unknown dynamics. In this case the assumption of Corollary 2 is made so as to fit the spectrum of the observed actual brain wave. For details we refer to [5]. Example 2. Stochastic area due to Paul Levy. Let (Bi(t),B2(t)), 0 < t < 1, be the standard two-dimensional Brownian motion. P. Levy defined the area enclosed by the Brownian trajectory and its chord within a certain time interval, say [0,1]. His expression is 5(1) = ^ / [£ 2 (<)dBi(t) - B1(t)dB2(t)}. * Jo We understand it as a stochastic integral. In the space Hi, we find versions of B\{t) and B2{t) as follows: Bi(t) = (x,X[o,t]),

B2{t) = (x,X[-t,o]>-

With these versions we can easily see that 5(1) is realized in the space H2 and have the kernel F of the integral representation. In fact, F is a function as shown in the diagram. It is an example satisfying property (22) of Corollary 2.

19 The characteristic function and semi-invariants are given by x(z)=(cosh|)

,

72 P +i = 0,

72P =

(4p - 1) -Bp,

Bp is Bernoulli number,

(respectively see [2], [7]).

§5. Prediction for the square of a Gaussian process The results discussed in this section have been obtained by G. Kallianpur and the author [23], in which the integral representation of an % 2 -functional is used. Let us begin with a brief general remark on prediction problem for a second order stochastic process {X(t)}. Having observed {X(u); u < s} we wish to predict X(t) for future t, i.e. t > s. Error is usually evaluated as a mean square error. Denote by X(t; s) the predictor for a future value. Then X(t; s) should be a "function of {X(u);u < s}" = ip(X(u);u < s,t). The best predictor means the one that minimizes the quantity (23)

E{X(t)-X{t;s)}2.

e(t;s) =

There have been two approaches: 1) Linear prediction where

ei(t;s)>en(t;s),

t>s.

Three remarks are now in order. i) The best nonlinear predictor is nothing but the conditional expectation: (25)

Xn(t;s)=E(X(t)\Bs(X)),

where BS(X) is the cr-field generated by the X(u),u < s. On the other hand, the linear predictor is obtained by taking an orthogonal projection of X(t) to the subspace of L2(Q.,p) generated by the X(u), u < s. ii) For a Gaussian process we always have Xn(t,s)

=

Xl(t,s),

since the conditional expectation E(X(t)\Bs(X)) of X(u), u < s.

is expressed as a linear functional

20 iii) A simple example of nonlinear prediction is a martingale. So, there is some advantage to express a given stochastic process in terms of martingale. Suppose we are given an JV-ple Markov Gaussian process {X(t); t > 0} with E[X(t)} = 0. Such an X{t) is defined to be a Gaussian process satisfying (26)

LtX(t) = B(t), X(0) = 0,

where Lt is an ordinary differential operator of order N with C^-class coefficients. Then, we have the canonical representation: (27)

X(t)=

f

F(t,u)dB( u)

Jo

satisfying (28)

E{X{t)\Bt{X))

= I F(t,u)dB{u),

t>s.

Jo

The kernel F(t, u) can be taken to be Green's function of Lt- More precisely, F is a Goursat kernel of Volterra type expressed in the form N

(29)

F(t,u)=> 0,

u >t

where {/i,..., /AT}, ({<7I, • • • ,9N}) forms a fundamental system of solutions of the differential equation Ltf — 0 (£*g = 0, L*u being the adjoint operator for Lt), and where d dN~2 (30) F(t, t) = -F(t, u)\u=t = •••= gp^Ht, «)l«=t = 0. Such an JV-ple Markov Gaussian process {X(t)} is realized in our space Hi by taking B(t) = {x,X[o,t]), t>0, and by using the representation (27). We are now ready to discuss nonlinear prediction of a stochastic process {Y(t); t > 0} defined by (31)

Y(t) = X(tf

- E(X(t)2),

t>0,

which always lives in I-L2 • The integral representation tells us that a kernel F(t, u)F(t, v) (e L2(R2)) is associated with Y(t). It is written as 53/»(*)/j(*)ffi,j(w,«;0. i>j

where giJ(u,v;t)

= ( 1 - -5itj

) {9i{u)gj{v) +

gj(u)gi{v))x[o^{^,v).

21 It is noted that a stochastic process Mij(t) with gij(u,v;t) as a kernel of the integral representation has to be a martingale (in % ) relative to the increasing CT-field Bt(B). Thus Y(t) is a linear combination of ^N(N + 1) such martingales:

(32)

r(0 = ^/i(i)/J(0MJ(i).

Proposition 5.1. Set Qx(t) = 6{X^(t)X(k\t)

-

Uk(t);

0
where X{j)(t) = (£t)jX{t) and lj>k{t) = E{X^ linear subspace of (L2) spanned by {• • • }). Set nY(t)

= o-{rational functions

(t)X^

of Y^(t);

{t)). (
0
l}f]H2,

then (33)

Qx{t)

C TlY{t).

Proof. To prove that J ( J ' ' ( t ) l W ( t ) - 7j]fc (i) belongs to TlY{t), we use induction with respect to j , k. First X(t)2 — 7o,o(£) € 7£y (*)• Next assume that XW(t)X<-k\t)

-

7jifc (t)

e TZY(t),

for k,

l
Then we see that all the following live in lZY(t): a) X H ^ p W ^ - v u M

[X^(t)X(k\t)-^n>k(t)}

= £

-[lW(i)X(W)(t)-Xw(t)], b) X("+D(*)*(")(i) - 7 n + l , „ ( t ) = ^

[ * ( " W - 7 n,n(*)] ,

C) X ( " + D ( i ) 2 - y n + l , n + 1 ( t ) = { i | [ X W ( t ) —

2

- 7n,n(*)] + 7 n + l , n W } 2 A ( n ) W 2

7n+l,n+l(^)-

Thus the assertion is proved. P r o p o s i t i o n 5.2. independent.

For eac/i t, iAe martingales Mij(t),

i > j , are linearly

For proof we use the integral representation of Mij(t) and the fact that 5j(w)'s are linearly independent in L 2 ([0, t}) for any t > 0. Details are omitted. Proposition 5.3. Set m(t) = 6{Mij(t); 1 < j < * < iV}. Then we have (34)

m(t) =

Qx(t).

22 Proof. By the canonical property of the representation (27) we know that each Gaussian martingale JQ gi{u)dB{u) can be expressed as a linear functional of the X(k\t)'s, 0 < k < N — 1. Now observe the integral representation of a member of Qx(t)- It has to be a linear combination of {gi(u)gj(v) + gj(u)gi(v)}x[o,t]2{u>v)The same holds for a member of m(i). Thus the equality follows. Corollary 1. Each martingale Mij(t) k
0 <

=$ij(Y(t),Y,(t),...,Y(N-1\t)-t).

Mi)0(t)

Corollary 2. Each {Mij(t);t

is a rational function of the Yk(t)'s,

> 0} is a martingale relative to Bt(Y) (Cjj Bt(B)).

Proof. For s
= E(E(Mid(t)\Bs(B))\Bs(Y)) = E{Mitj(s)\B.(Y)) = Mij(s) (by Corollary 1).

Theorem 7. The best nonlinear predictor for Y(t) is given by (36)

Yn(t,s) =

Y^fi(t)fj(t)$ij(Y(s))...,Y(N-1\s);s),

where $ y is a rational function with parameter s given by (35). Proof. We recall that Yn(t,s) = E(Y(t)\Bs(Y) expression (34).

and use Corollary 2 to obtain the

Remark. One may ask if there is a significant difference between Yn{t, s) just obtained and the best linear predictor. Concerning the error we know that the inequality (24) holds in general. In our case, if t is fixed, en(t, s) and e;(i, s) behave as is shown in the following figure, although we have not succeeded in finding an explicit expression.

I

^-^1 t

>

s

Example. Numerical values of e„ and e; as well as the best linear predictor can be obtained in the case where X(t) is given by X(t) = [ (t- u)dB(u), Jo

(double Markov).

23 The Y(t), being viewed as a second order stochastic process, is expressed in the form Y(t) = / {at2 + btu - (a + b)u2)^d£(u), Jo where a = (%/l9— l)/\/3, b = - ( 2 \ / l 9 - 4)/\/3, and d£(u) is an orthogonal random measure which is causally related to Y(t). We therefore have Yi(t,s)=

/ (at2 + btu-(a Jo

+

b)u2)Vud£(u).

It is somewhat surprising that d£(u), 0 < u < s, and Yi(t,s) is a linear functional of all the y(w)'s, u < s, while Yn(t, s) is obtained from the germ of Y at s as was seen in Theorem 7. §6. Infinite dimensional rotation group We shall make use of the infinite dimensional rotation group introduced by H. Yoshizawa [16] to investigate the measure \x of W.N. with the aim of developing harmonic analysis on E*. There were several motivations for us to use this group. Among them the following are worth mentioning. a) Structure of the "support" of fx. It is somewhat awkward to speak of the support of fi, however it might give an idea if we identify a class of transformations acting on E* under which fi is invariant. Let us recall a technique used in the proof of Proposition 1.3. With a complete orthonormal system {£ n } in L2(R) we formed a sequence xn = (£ n ), n = 1,2,..., of independent standard Gaussian random variables. The {xn} may be thought of as the coordinates of x g E*. The strong law of large numbers suggests us to have the following intuitive observation: supp(^)

f

i

N

}

approximated by < x; — ] P x2n = 1 \ Roughly speaking, supp(/i) looks like a limit of spheres, which is invariant under the rotations. b) Further naive observation might be given in line with a). The measure /x seems to be the limit of the uniform probability measures on spheres. Such a uniform measure can be characterized as a rotation invariant measure. Similar characterization of fi might be possible by introducing a rotation of E*. c) The so-called flow of the Brownian motion is a one-parameter group of /li-preserving transformations on E*. There would be no need to emphasize the importance of the flow. We wish to find much bigger group involving the flow of the Brownian motion. We now come to the definition of the infinite dimensional rotation group. The basic nuclear space E is taken to be S or Do- The collection of linear transformations g satisfying

24 i) g is an isomorphism of E, ii) ||
(ffiS2)£ = fli(520-

With this product the collection 0(E) forms a group. Definition. The group O(E) is called the infinite dimensional rotation group, and each member g of 0(E) is called a rotation of E. When E is not specified, the group is denoted by O(oo). Remark. The group 0(oo) can be topologized. Indeed, we introduce either 1) compact-open topology, or 2) iz-topology (use the operator norm of (Ug : (Ug
(x,gO = (x*,0The mapping x ->• x* defines a linear isomorphism g*. Write x =g x and 0*{E*) =

{g*;geO{E)}.

A product is introduced in 0*(E*) so that it is in agreement with that in 0(E): (38)

9*i9*2 =

(9291)*.

Proposition 6.1. With the product (38) 0*(E*) forms a group. It is isomorphic to O(E) under the correspondence

The most important, and in fact expected, property of O(E) or 0*(E*) appears in the following. Theorem 8. i) Each g* G 0*(E*) is B-measurable, where B is the a-field generated by cylinder sets. ii) g*n = /x, for every g* G 0*(E*).

25 Proof, i) is immediate because g* carries a cylinder set to another cylinder set. For ii) we compute /

exp[i(x,£)}d(g*fj,)(x) = /

exp[i(x,£)]dfi(g*x)

= I expli^*- 1 ^, 0]Mv)

(V = 9*x)

= I exp[i(y, g-lt)}d»(y)

(s*"1 = (ff)"1*)

and use Theorem 1 to obtain the conclusion. The next step is to discuss some subgroups of 0*(E*). Since 0*(E*) is isomorphic to O(E), we discuss on O(E) which is somewhat easier to deal with. [I] Finite dimensional rotations. Take a finite number of linearly independent vectors, say £ 1 , . . . ,£„. Without loss of generality it is assumed that they are orthonormal. Any £ is expressed in the form n

£ = X ! ak^k + £''

£' i s o r t hogonal to £ 1 , . . . , £„.

fc=i

Let gn be a member of SO(n), and let

G)-G) Then a transformation g defined by n

6 < + £' fc=i

belongs to O(E). Thus we see that for every n, O(E) has a subgroup Gn isomorphic to SO(n). One might expect that the limit of Gn would cover the whole group O(E), however this is not true in reality. We can find quite different kind of subgroups of O(E), as is shown in [II]. [II] Time change of W.N. Consider a transformation g^, of £ defined by

(39)

9*:t(u)^ZMu))VWW\,

26

where ip is some monotone mapping of R onto itself. Once E is specified, we can discuss conditions on ip under which g^ belongs to 0(E). Moreover, we are interested in a one-parameter subgroup {gt} consisting of transformations expressed in the form (38). In such a case {gt} is determined by a system of monotone functions {il>t(u)} satisfying (40)

4>t(Mu))=A+s(u).

A typical and most important example of {gt} is the shift {St}. It is defined by (41)

St : £(u) -+ £(u - t).

The adjoint for St is written as Tt(= Sf). By Theorem 8 each Tt is a measure preserving transformation acting on (E* ,fi), and {Tt} forms a one-parameter group. Thus we are given a flow on (E*,n). Definition. The flow {Tt} is called the flow of Brownian motion, (S. Kakutani). Remark. In terms of B the Tt acts in such a way that Tt:B(t)

_ • £ ( « + *).

With the flow {Tt} a one-parameter group {Ut} of operators on (L2) is associated in such a way that (Ut

Since Tt is measure preserving, {Ut} is a unitary group, and it is continuous in t. We therefore have the spectral decomposition {Ut} (Stone's theorem):

Ut = I exp[it\]dE(X), where {E(X); —oo < A < oo} is a resolution of the identity. We can also speak of the spectral type. Note that {Ut} on (L2) is reduced to T-in, n> 0. Theorem 9. The spectral type of {Ut} (or of the flow {Tt}) is simple Lebesgue on T-Li and is countably Lebesgue on %n, n > 2. Proof. Denote by i the isomorphism ol/Hn and L2(Rn) i(Hn) = LHR^). Set iUti-1 = Vt. Then VtF(ui,...,un)

= F(ui

established by Theorem 5:

-t,...,un-t).

Since {Ut} and {V^} are unitarily equivalent, the question is now to investigate the spectral type of {Vt}. In case n = 1, {Vt} is simply the shift on L2(R1), and hence the spectral type is simple Lebesgue. For the case n = 2, we first restrict F(u\,U2) to the half-plane ui > Ui and use two complete orthonormal systems {£„} and {r]n} in L2(R), and i 2 ([0,oo)) respectively, to obtain the expansion (42)

F(uuu2)

=^2aJ
0 f ' o "

2

Vk{ui-u2).

27 Then we see that Vt acting on F leaves r/k invariant. If £,-, the Fourier transform of £j, does not vanish a.e., then (42) shows that L2{R2) can be decomposed into countably many (as many as Jfa's) cyclic subspaces and that on each cyclic subspace {Vt} has simple Lebesgue spectrum. Thus we have proved that {Ut} on H2 has countable Lebesgue spectrum. We use a similar but slightly more complicated technique for {Ut} to have countable Lebesgue spectrum on %n, n > 3. Addenda 1) let Bt be the cr-field generated by the {x,£), supp(£) C (-00, t\. The system {Bt]t € R} satisfies i) Bt D Bs for s < t, ii) C\Bt = {hE*},

mod 0,

in) VBt = B, mod 0. In connection with {Tt} we have iv)

TtBs=Bt+s.

Because of i)-iv), Tt is a Kolmogorov flow. A general theory of a dynamical system asserts that the spectral type of a Kolmogorov flow is countably Lebesgue. Hence Theorem 9 has been almost proved, however the trick of the above proof tells us something more. For instance, it suggests the algorithm to form Weiner's nonlinear network system shown in his book [5] (also, see §0, Example 2)). 2) The integral representation of a martingale adapted to the family of increasing cr-field Bt defined above has a special property. Proposition 6.2. Let X(t) be a martingale adapted to {Bt}, and assume that it is always living in rln for some n. Then, there is a function F{u\,..., un) which is square integrable on any domain (—oo,t] n , and the kernel of the integral representation of X(t) is given by F(ui, ...,un)=

X(-oo,i]« F(ux,...,

un).

§7. Projective invariance of Brownian motion We are still, in this section, in search of one-parameter subgroups of O(E) belonging to the class (II) in §6. Before we come to this main question let us have a short observation of the projective invariance of Brownian motion. Consider all the trajectories of a Brownian motion starting from the origin and coming back to the origin at a fixed time, say t = 1. This is called a Brownian bridge or a pinned Brownian motion; call this process X(t), 0 < t < 1. Set

Y(t) =

v/Var(X(i))'

0 < t < 1,

28

and let tp be a projective transformation of (0, 1) onto itself. Then {Y(t); 0 < t < 1} and {Y((p(t));0 < t < 1} are the same Gaussian process (P. Levy obtained this property in more general case). Such a property of Brownian motion is called the projective invariance. It gives us some deep insight of properties of a Brownian motion. In what follows we shall show that the projective invariance can be explained in terms of a subgroup of O(E) being in line with §6, [II]. Now that importance of {Tt} (or {St}) has been discovered in §6, one should keep {St} in mind and take an exploring route to find other one-parameter subgroups of 0(E) which have probabilistic meanings. This can be done by using infinitesimal generators. With some reasonable assumptions, ipt(u) satisfying the relation (40) is expressed in the form (43)

ipt(u) = f(f~l(u)+t),

/ : monotone.

We restrict the class of the ipt(u) as above, and define gt by

(44)

Gfc£)(«)=£(V*(«))i

1MU)

Assuming that / in (43) is piecewise smooth, we obtain the infinitesimal generator a = -^j.gt\t=o- It is expressed in the form (45)

a = a(u)— +

-a'(u),

where a(u) = / ' ( / _ 1 ( u ) ) . For example, the shift {St} has the generator s = — ^ . In terms of generators, we observe possible relations between s and general a. 1°) [a, s] — 0, where [ , ] means the Lie product. (46)

[a,s}=a'(u)^-

+

^a"(u).

This shows that a = cs + c', c, c' constants, so no new generator arises. 2°)

[a, s] = Xs,

A constant.

In this case a = — X-4— k up to linear function of s. Take A = 1, and one obtains a one-parameter group {rt}: (47)

(7tO(«)=£(ueV/2-

The one-parameter group {rt} is called the dilation (or tension). Its generator, call it r, is d 1 The commutation relation [r, s] — —s shows that (49)

StTs=TsSte,,

29 namely {St} is transversal to {r} as is illustrated by the picture.

Ste'

^

Now we look for the third generator, again denoted by a. Noting (50)

[a, r] = (ua'(u) - a(u))—

+

-ua"(u),

we consider the following possible cases: l°)[r,a]=0, 2°) [r, a] = As, A constant, 3°) [r, a] — fir, fi constant, 4°) [T,O\ = 7a, 7 constant. From l°)-3°), no new generator is given, while the relation 4°) gives us 7+i

d du

7+ I 2

l)(«^

+

With this a we have (see (46)) [a, a ] = ( 7

+

2u7-i).

Take 7 = 1, and set 2 ^

= u - — h u. Then S{s, r, K} forms a Lie algebra. K

Remark. If 7 > 2, no finite dimensional Lie algebra involving s, r, K is obtained. The K generates a one-parameter group {Kt} given by (51)

(KtO(u)=*'

" -tu+lj

^

1

I - t u + l|

Now we must specify the muclear space E to be Do so that {St}, specifically {«t} are subgroups of 0(DQ).

{rt} and

Theorem 10. With the choice E = D0, {St}, {rt} and {nt} are one-parameter subgroups of O(DQ). Their generators satisfy [S,T]

(52)

=S,

[r, K] = «, [K, S]

= 2r.

30 Having observed the commutation relations (52), we are led to find a connection with the group SL(2,R). In fact, we have the following observation: Let ip be in GL(2,R). Consider a mapping

(53)

^U^^7^

*={c c*J

If E = Do, then gv G O(D0). Furthermore, the above mapping is a homomorphism of GL(2,i?) into 0{DQ), and the kernel is the center of GL(2,i?). Denote by Gp the subgroup of 0(DQ) generated by {St}, {rt} and {K;J}. Then we have an isomorphism PGL(2, R) ^ Gp. We are now in a position to give a probabilistic interpretation to these three one-parameter groups. The shift has already been discussed in §6. For the dilation {rt}> w e see, through the relation (49), that {Tt} is a transversal flow to {T4*} on (DQ,fi). By using {r*} we define a stationary Gaussian process U(t) by U(t)=U(t,x)

= {T?x,X[o,i}).

It is in T-Li and has covariance function exp[— ^\h\). This shows that {£/(£)} is the Ornstein-Uhlenbeck process. Definition. The flow {rt*} is called the flow of the Ornstein-Uhlenbeck process. The spectral type of {rt*} is countably Lebesgue: on Hi it has double multiplicity and on %n, n>2, countable. We then come to the third one-parameter group {/tt}. With the help of {St} and {rt}, the group {« t } describes the projective invariance of Brownian motion. Fix an interval (a, b) and define

f(u'f) =

b— t

Y^T^XM( U )

y/b — a

(We come to the expression from the canonical representation of the Brownian bridge.) Let ipbe a, projective transformation of (a, b) onto itself such that
g*f{-;t) =

f{-;V-l{t)).

With this relation in mind we can immediately prove Proposition 7.1. Two stochastic processes given by {(x,f(-;t));a
31 Remark. We can illustrate the projective invariance in more general setup as was done by Levy. For example, we can take p such that p(a) = b, ip(b) = a; or take

§8. Some applications of O(oo) Characterization of the measure /x of W.N. was one of the motivations to use the group O(oo). Interesting work on this topic has been done by Y. Umemura [11] and others. Some of those results will be described in this section. To fix ideas we assume that E = S. Let {Tt} be the flow of the Brownian motion. Theorem 9 implies that fj, is T t -ergodic. Since {Tt} is a subgroup of 0*(S*), we see that /J is 0*(5*)-ergodic, namely any 0*(5*)-invariant set is either ^z-measure 0 or 1. A somewhat stronger assertion can be obtained. Set Gf = {g G O(S); g is a finite dimensional rotation}. This is a group (in algebraic sense). Proposition 8.1. The measure /J, is Gf-ergodic. Proof. It suffices to show that "If for a real bounded functional {g*x)) =
=
Since Bn approaches B, cp^ —> 0. The same for Ug 0. N ° w w e choose a g £ Gf such that {g£i,---,g£n}

J- {6.••-.&*}•

Then
2E{pn)E{Ugpn)

2

= 2[E(

32 Theorem 11. (Umemura) Let v be a probability measure on (S*,B). Ifv is 0*(S*)invariant (g*v = v for any g* S 0*(S*)), then (54)

v = aSo + /

/iCTdm(cr),

a > 0,

J(0,oo)

where 6Q is the delta measure sitting on the origin and\ia is the measure of W.N. with variance a2, and where m is a Borel measure on (0, oo). Remark. The representation (54) is in fact a necessary and sufficient condition for v to be 0*(5*)-invariant. The proof of Theorem 11 uses the following two lemmas. The first one is easy, while the second one is well known. Lemma 1. If v is 0*(S*)-invariant, function of ||£|| : Cv{§ = f(U\\)-

then the characteristic functional of v is a

Lemma 2. (Bernstein) If f(t), t > 0, is completely monotonic: for every n,

J2(-l)kffjf(t

+ ka)>0, t>0,

a>0,

then there exists a Borel measure m on [0, oo) such that POO

(55)

f(t) = / Jo

exp[-Xt]dm(X).

Idea of the proof of Theorem 11 1) Using Lemma 1, one has

CA0=f(U2\\),

/on[0,oo).

2) To prove that the above / is completely monotonic one follows the following three steps. « /(*) > 0. Let { f i , . . . ,£„} be an orthonormal system in L2(R), is positive definite,

The left-hand side is n + n(n — l)f(t). non-negative.

and set £, = \j\t,j-

Since Cu

This is true for every n, so f(t) must be

ii) Set / = /o and set /i(||£|| 2 ) = -/o(||£|| 2 + a) + /o(||£|| 2 ), « > 0, and prove that h is positive definite. Similarly we define / 2 , / 3 , . . . and prove that they are positive definite. Thus / is complete monotonic.

33

iii) We have, via Bernstein's lemma /•OO

CAO = /

exp

•£l«ll»

dm(a),

Jo

m({0}) + f Jo

Ht

exp

dm(a).

This gives the representation (54) of the measure v. Corollary. If O*(S*)-invariant probability measure v on (S*,B) is 0*(S*)-ergodic, then v is either So or fia with some a > 0. Addenda 1) With a choice of a complete orthonormal system {£„} in L2(R) we may define the infinite dimensional Laplace-Beltrami operator A^,: A0

= £ e&

d {x n)

^

It enjoys the properties i) a quadratic form of generators of one-parameter subgroups of O(S), ii) commutes with O(S), iii) negative, iv) annihilates constants. Futhermore, the Wn becomes the eigenspace of AQO : AooV? = -rup,

Un.

These observations indicate a close analogy to the sphere Sn = SO(n + l)/SO(n) with uniform (rotation-invariant) measure. Note that the direct sum decomposition of (L2) given by (12) might be viewed as an infinite dimensional analogue of the decomposition of L2(Sn) into subspaces, each of which is spanned by the spherical harmonics with the same degree. 2) Having observed the analogy discussed above as well as the motivation a) in §6, one can think of a comparison of the classes [I] and [II] in §6. The collection of finite dimensional rotations plays an important role in the discussions in this section and further we know that AQQ can be determined by it. On the other hand, these "whiskers" (see diagram) play somewhat different roles as we have seen in §7. We can even discriminate between these two classes by, say, average power. It would be an interesting problem to discover another whisker that describes some important properties of W.N.

34

§9. Generalized Brownian functionals We wish to analyze Brownian functionals by using the integral representation, where we are naturally led to generalize Brownian functionals. Before discussing differential calculus of functionals t/(£) on S, we introduce various notations and definitions following P. Levy [3]. 1) Variation 6U(£) of a functional £/(£) (in the Frechet sense). Set At/(£) = U(£ + <50 - U(£). 6U(£) is said to be the variation of U if a) it is a continuous linear functional of 6£, b) Af/(0 - 6U(£) = o(5£) as 6£ -> 0. If SU exists, then 5U(Q =]im

\{U{Z +

\5£)-U{Z)}.

(variation in the Gateaux sense). 2) Functional derivative Since SU is continuous in 5£, it is expressed in the form 6U{t) = {U'v6£),

U'^eS*.

If U't is an ordinary function, then it is called the functional derivative of U and is denoted by U^,t). 3) Entire functional of degree n. A functional U(£) is said to be an entire functional of degree n if U satisfies U(\£ + fir)) is a polynomial in A, /z of degree n. If, in particular, it is a homogeneous polynomial, then U is said to be entire and homogeneous of degree n. For n — 2, we simply say that it is quadratic. 4) Second variation 52U(£). If 82U(£) satisfies

35 a) S2U(£) is a quadratic function of 5£, b) MJ - (6U + \52U) = 0(8?) as <5£ -> 0, then it is called the second variation of U. 5) Functional derivatives of the second order If 5U^ is an ordinary function of t and if it can be expressed as

dU'^t) = U£(t)6t(t) + Ju&l(t,s)6S(8)ds, thenU'J-iit) and U'Jf,{t,s) are called functional derivatives of the second order. 6) Laplacian A/, in Levy's sense. It is defined by

If

ALU(£)

— 0, then U is harmonic.

Several examples are going to be presented. The computations of variations, functional derivatives and Laplacian are straightforward, and there is no need to present them here. However, each example represents a Brownian functional or a generalized Brownian functional through the integral representation. Example 1. J7i(0 = ff £(u)£(v)d[i(u,v), fi : symmetric measure. If n is absolutely continuous, then U\ is said to be regular. A regular functional can be represented as a kernel of the integral representation of %2-functionals. The regularity can easily be extended to the entire, homogeneous functional of higher degree. Example 2. U2(0 = J f(u)£(u)2du. This is an entire, homogeneous functional of degree 2. If £/(£) is expressed in the form

^(0 = ^1(0 + ^ ( 0 , then U is called normal. A class of normal functionals seems to be a basic class for which functional analysis would be well developed. Example 3. U3(£) =

Jg(u)£(u)3du.

Example 4. t / 4 ( 0 = / £ ( " ) £ ( ! - u )*xThis is a special case of the functional in Example 1. Example 5. U5(£) = f

h(u,v)£(u)£(v)2dudv.

Example 6. C/6(0 = / / k(u, v)£(u)2£(v)2dudv. We see that each of the functionals shown above is entire, homogeneous and that a regular U\ or U2 is harmonic if J f(u)du = 0, and so forth.

36

Let us recall that a functional in J-n can be expressed as (Theorem 5) inC(£) J---J

(56)

F(Ul,...,

where F € L ^ R " ) . Write it as inC(£)U(£). Introduce the notation

(57)

un)Z(Ul)

• • • i{un)dUl

• • • dun,

This £/(£) is regular and homogeneous of degree n.

^ = TOW(^(Oe^},

and we have an isomorphism (58)

Un S

fn

via the transformation T and by (57). Now take the functional U in J-n. Then its functional derivative U'At) exists and it can be expressed as (59)

U'z(t) =nj

R;1I

J F(t, u2,...,

un)Z(u2) • • •

tfu^du"-1.

This is a functional living in J-n-\ with parameter t. It is associated, by the isomorphism (58), with a stochastic process living in /Kn-\We then come to the functional derivatives of the second order: (60)

(U»4a{t)=0

<^

[ U^ (t, s) = n(n - 1) / • fRn_2 F(t, s,u3,...,

un)Z(u3) • • • Z{un)dun~2.

It follows that [/(£) is harmonic: ALU(£) = 0. There is one important remark. In the above discussion one assumption is missing. Namely, U'At) may not be defined for each t, since F is only an L2{Rn)function, and similar to those in (60). Thus it seems to be quite reasonable to restrict the class of F to tf("+i)/2(R") = tf(™+1)/2(#n) n L 5 ^ ) , where ff("+1)/2(JR") is the Sobolev space of order (n + l ) / 2 on i? n . Define

^n)

=

UnC(0J-jF(u1,...,un)^u1).-.^un)dun;FeH^^^(Rn)\.

By analogy with (57) we can define F^. Introduce the #( n+1 >/ 2 (.R n )-topology on Tn\ i.e. if inC{£)U(£) is in Fn with kernel F, then the norm of it is denned to be the iJ( n+1 >/ 2 ( J R n )-norm of F times y/ri. (cf. Theorem 5ii)). Set H^ = r _ 1 ( ^ n ) ) and topologize it so that T is an isometry. Let Tin and ^ r ( _ n ) be the dual space for Hn and T^, respectively. Then we have the following diagram: T(n) r T J n

> J n

$

$

r -£•(-") > •J n

$

37 The vertical arrow means isomorphism and C> means continuous injection. Of course, Fn , F and .T^ - "' in the diagram can be replaced by Tn , Fn and Tn , respectively. Definition. degree n.

A member of Tin

is called a generalized Brownian functional of

We now return to the examples of generalized functionals. For Ui{£), it in the form

we

express

U2(0 = JJS{U - v)f ( j ^ j i{u)i{v)dudv. The kernel is in H-3/2(R2), and it is approximated by F„(u,v) = 12i iX&?(u,v), where A» = [^,^p] and a* = nfAf(u)du. The Fn tends to na

the kernel of U% in i?~ 3 / 2 (i? 2 ). This means U2 € ^ - 1 ) . Thus U2 can be associated with a generalized Brownian functional. Although such a functional cannot be expressed in the usual classical form, we wish to have an intuitive, formal expression. Since Fn is associated with a Brownian functional ^2inai{(AiB)2 — |Aj|} = S i a i { ( ^ f ) 2 - JK~\} • |A»li the kernel 5{u - v)f(y^-) would be associated with a generalized Brownian functional with a symbolical expression (61)

2J

f(u)H2(B(u);jAdu,

where the Hermite polynomial with parameter used: Hn(x;a2)

=

. H,

!2"/2""VV2
For Us(£) we see that it belongs to !Fz, and we can play almost the same game to find a generalized Brownian functional that is associated with t/3 and has the following symbolical expression (62)

Jg(u)H3(B(u);jAdu.

All other examples except Ui(£) determine their own interesting generalized Brownian functionals. Details are left to the reader.

§10. Epilogue In closing this series of lectures I wish to present some concluding remarks and problems toward a proposed framework on the analysis of Brownian functionals. 1) Causal calculus The first topic is concerned with the choice of a coordinate system fitting for the causal calculus. In order to analyze the member of (L2), we introduced polynomials and a basis consisting of Fourier-Hermite polynomials, where {(x,£n)} with a

38 complete orthonormal system {£„} in L2(R) plays a role of a coordinate in E*. This is quite convenient from the standpoint of what we have discussed in §8, in particular, Addenda 1). However, when the propagation of time is taken into account, {B(t); t G R} seems to be more reasonable, although the number of the coordinates is continuum. Indeed, {B(t)} has independent values at every t and further Tt carries B(s) to B(s +1). In line with this idea, symbolical expressions such as (61) and (62) are very suggestive. A related calculus is the multiplication by B{t). This can be done only in the space of generalized Brownian functionals (L2)~ (the space involving all the Tin functional). The following example is suggestive. Example. U(£) = £(t0) J f(u)£,(u)du, t0 fixed, / smooth. This is in ^ • Again, we can associate a generalized Brownian functional:

B(t0) J f(u)dB(u) - f(t) with the U(£). It is not difficult to generalize the above example. Indeed, the product B(t0)Hn (^?§M, V € S, \\T)\\ = 1, does exist and belongs to ?4i+™ + Hn-iThe product has integral representation ( T can be extended over the n(nrn)) of the form z"+1C(e)2"/2{^(io)(e^)n-1+^(io)(?,77)"-1}. We are now ready to establish the product B(to)tp(x),

dB(t)-

•

v(n) n

_> V

1

"- 1 -

This has been established in terms of J-n by using the functional derivative (see (59)). It is interesting to develop the analysis including this causal calculus. 2) Application to quantum mechanics Let us come back to Feynman's path integral that was briefly mentioned in §0.3). A propagator tp would be obtained by the integral ip(x,t)

/ exp i— I x(s)2ds

exp —i I

Vds Vx.

Once x (possible trajectory) is thought of as a sample function of a Markov process, we must meet a difficulty to explain J0 x(s)2ds. If x(s) behaves like a sample path of W.N., i.e. like B(t), then we know how to deal with it. Thus if the factor coming from the potential is well defined (call it (L2)+) the path integral

39 will make sense. We have hopes that this will develop into a physically acceptable theory. 3) On the search for the Fourier transform So far three different kinds of Fourier transform on (L 2 ) have been considered, but none is satisfactory. a) The transformation T. It was given by ( 7 » ( £ ) = / ex-p[i{x,£)]

and satisfies
for

f&Un.

2

This is a beautiful transformation on (L ), however it does not seem to be appropriate for the causal calculus. c) Another Fourier transform. Start with

40 References [1] T. Hida, Stationary Stochastic Processes (Princeton Univ. Press, 1970). [2] T. Hida, Brownian Motion (Iwanami Pub. Co., 1975) (in Japanese). [3] P. Levy, Problemes Concrets d'analyse Funtionnelle (Gauthier-Villars, 1951). [4] H.P. McKean, Stochastic Integrals (Academic Press, 1969). [5] H.N. Wiener, Nonlinear Problems in Random Theory (Tech. Press of MIT, 1958). W h i t e noise, nonlinear functionals [6] R. Cameron and W.T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Ann. Math. (2) 48 (1947) 385-392. [7] T. Hida, Quadratic functional of Brownian motion, J. Multivariate Anal. 1 (1971) 58-69. [8] T. Hida and N. Ikeda, Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral, Proc. 5th Berkeley Symp., Vol. 2, part 1 (1967) 117-143. [9] K. Ito, Multiple Wiener integral, J. Math. Soc. Japan 3 (1951) 157-169. [10] P. Levy, Le mouvement brownien, Mem. Sci. Math. CXXVI 1954).

(Gauthier-Villars,

[11] Y. Umemura, Measures on infinite dimensional vector spaces, Pub. Res. Inst. Math. Sci. Kyoto Univ. A l (1966) 1-47. [12] N. Wiener, Homogeneous chaos, Amer. J. Math. 60 (1938) 897-936. Orthogonal or unitary group [13] T. Hida, Complex white noise and infinite dimensional unitary group, Lecture Note (Nagoya Univ., 1971). [14] T. Hida, A role of Fourier transform in the theory of infinite dimensional unitary group, J. Math. Kyoto Univ. 13 (1973) 203-212. [15] T. Hida, I. Kubo, H. Nomoto and H. Yoshizawa, Projective invariance of Brownian motion, Pub. R.I.M.S. Kyoto Univ. A 4 (1968) 595-609. [16] H. Yoshizawa, Rotation group of Hilbert space and its application to Brownian motion, Proc. Int. Conf. on Functional Analysis and Related Topics, Tokyo (1970) 414-423.

41 Generalized functional of Brownian motion [17] T. Hida, Analysis of Brownian functional, Proc. Lexington Conference on Stochastic Systems, 1975. [18] T. Hida, Functionals of Brownian motion, Proc. VII Prague Conference on Information Theory, 1974. [19] P. Levy, 1951 book. (loc. cit.) [20] J.L. Lions and E. Magenes, Nonhomogemeous Boundary Value Problems and Applications, I. (Springer-Verlag, 1972). Applications and Motivations [21] A.V. Balakrishnan, Stochastic bilinear partial differential equations, Proc. U.S.-Italy Conference on Variable Structure Systems, Oregon, 1974. [22] D.A. Dawson, Stochastic evolution equations and related measure processes, J. Multivariate Anal. 5 (1975) 1-52. [23] T. Hida and G. Kallianpur, The square of a Gaussian process and non-linear prediction, J. Multivariate Anal., 1975.

42

A d d e n d a (from the second edition, 1978) §11. Generalized Brownian functionals (continued) There were several motivations, as we explained in §0 as well as at the beginning of §9, in introducing the concept of generalized Brownian functionals. Recall also that we briefly mentioned in §10, 1) causal calculus, where the passage of time is involved explicitly in terms of B(t). With these topics in mind we shall reconsider how to generalize Brownian functionals. The causal calculus which was proposed in §10 is taken into account in this development. Since B(t) is a formal expression, our starting point should be an approximation to B(t). Let {Afc} be a partition of R1 and denote by AkB the increment of Brownian motion over the time interval Afc. Now

m

approximates

{B(t)}

and the time shift induces the transition AfcB ->• Afc+iB. A functional of the B(t)'s can now be approximated by a function of AfcB's. Assuming that it has finite variance, we can say that the approximating function can be expressed as a linear combination of Hermite polynomials in AfcB's, since {Afc-B} is a Gaussian system. We therefore have a base consisting of Hermite polynomial for the L 2 -space of functionals of AkB's. For convenience, we take Hermite polynomials in (AfcB/Afc)'s and divide them into two parts:

(101)

class (I)

(102)

class (II)

J]

^ ,

Y[Hnk

( - r ^ ; j 4 - r ) , some nk > 1,

where Hn(x;a2) stands for the Hermite polynomial with parameter which was introduced in §9. Now let the partition {Afc} be finer and finer. Then we are led to formal expressions of the limits of (101) and (102). They are (103)

(104) respectively.

n^

\\Hnk

f c

)'

\^^'dr) '

V s different,

tk S

'

different

'

s o m e nk

> 1

'

43 Having made these observations, we now come to rigorous definition of the above formulas and their linear combinations (which turn out to be integrals) of them. Actually, this can be done within the framework of generalized Brownian functionals established in §9. We now start with our measure space (S*,fi) of white noise (W.N.). Let XA denote the indicator function of a set A. Then (x, XA) is a Gaussian random variable on (S*, fj.) and indeed it is a realization of AB on (S*,fi). A Hermite polynomial Hm{^-\ r^r) is therefore realized in the form Hm(^x'^A'; m ) . Now apply the transform T introduced by (3) in §3. Then, we have (105)

T

( *

m

=

( ( ^ )

;

^ ) )

( l )

iAim i g ( 0 / '„" \L\\mm\ J Rm J

XA">(ui,...,um)£(ui)...£(um)du1...dum.

Before we take the limit as |A| —> 0, we introduce a notation 5m{dum) : j

RmJ

f(u1,...,um)Sm(dum)=

J

J(u,...,u)du.

With this notation we see that the limit of i J m ( ' x ' ^ A ' ; r^r) in Hm may be expressed in the symbolic form Hm(x(t); ^ ) and that Hm(x(t); ^)dt = Mm(dt) plays a role of a random measure with the relation that T(Mm(dt)) = ^j5m{dtm). More general random measures are defined in the following manner: k

(106)

j Q *Mnj(dtj)

= Mni {dh) • Mn2(dt2)

• . . . • Mnk (dtk)

3= 1 (

k

JJM„.(^),

if t\,...,tk

are different,

j=i

. 0,

otherwise.

This can also be rigorously defined in a similar but complicated manner, and we can prove the following theorem. Theorem, i) For F G H(k+1V2(Rk), (107)

f = J -•

the integral

F(Ul,...,Uk)Mni(dUl)»...»Mnk(duk),

J2ni

=

n

is defined, and the integral ip belongs to Hurt) The integral representation of ip is given by

(108)

(7V>)(0 = j ^ j C(0 I -R- I F(uu ..., ufc)£(«i)ni ... £(Ufc)n* duk.

44 Remark. The transformation T in the expression (108) is the extension of T given by (5) in §3. This extension gives the isomorphism between Hn and J-fT • In view of the above theorem the product (106) is called a generalized random measure of degree n, and the integral of the form (107) is referred to as a generalized multiple Wiener integral of degree n. In case rij = 1 for all j the integral is in agreement with the multiple Wiener integral given in §3. Remark. The collection of all generalized multiple Wiener integrals of degree n does not cover the entire TuT • For some special cases we are able to establish multiplication formulas for generalized random measures. Mn(dt)M!(ds) (109)

Yl-M^idt,)

= Mn{dt) • Afi(ds) + J t _ a M n _i(dt),

Mi(ds) =

£ Mnj{dtj)Mx{ds)

Yl • Mni(dU)

These come from the multiplication formula for Hermite polynomials with parameter: min(m.ra)

TT ,

2MT /

2N

V^ fc=o

2J,

.

a^{m

.

+ n-2k)\

2

k\(m - k)\(n - k)\

By using the formulas in (109) we can get the product of some multiple Wiener integrals and generalized ones. For example,

J f(u)Mn(du) j g^M^dv) = I"I"(/ ® g)(u,v)Mn(du)

• Mi(dw) + J /(u)(7(u)M„_i(du),

where denotes the tensor product. Let Afn be the class of all 'Hn"n -functional which are expressed in the form (108), and set oo

J\f = 2~] A/"n

(algebraic sum).

71=0

A member of Af is called a normal functional. If, in particular, a normal functional is in % = 2 ^ = 0 ^ ™ ' ^ e n ft ^s c a u e ( ^ regular. Remark. The term "normal" and "regular" came from P. Levy [3]. In fact, after applying T to members of TV and H, then we are given up to the factor C(£), normal and regular functional, respectively, in the sense of P. Levy.

45 §12. Differential calculus The differential operator, whose symbolic notation is d dB(t)

or

d dx(t)

(see §10, 1)), has been defined by using the functional derivative explained in §9. Suppose that (p £ %n with the integral representation {T
r r U ( 0 = / — / F(m,...,

where F € L2(Rn). (111)

u n ) £ ( u i ) . . . £ ( u „ ) d « i . . . dun,

Then the functional derivative U^(t) is expressed in the form

U{(t) = n

— x / F(t, w2, • • •, un)£(u2)...

£,(un)du2 ... dun,

and the functional in~1C(£,)U'At) defines an %„_i-functional denoted by (112)

d~(Tf^ =

-Q^T^.


T-'^-'cw^t)).

Such a method to define a partial derivative by making use of the integral representation and the functional derivative is still applicable even when
(T
incW(0,

r r U(0 = J -R- J F{uu . . . , uk)i{Ulr

where J2j=i nj

= n an

d F € Hk(Rk).

• • • i{uk)nk

duk,

Then, we have

i=i

Applying T 1 to in 1C(^)U'^(t), we are given a normal functional in Mn-iWe then define a differential operator

(114)

Da= Ja(t)-^-dt,

aeL°°(R).

46 This smeared operator is sometimes easier to deal with rather than g^(t^ itself as is seen in the following theorem; however g ^ y is specifically important when the passage of time t should be expressed explicitly. Theorem. The vector space Af is in the domain of the differential operator Da, a G L°°(R), and each Da gives a surjection of J\fn onto Nn-i- In particular, regular functionals are carried to regular ones.

The differential operator ^ 4 ^ or the same thing

?

is nowadays denoted simply by dt.

Reprinted from JOURNAL OF MULTIVARIATE ANALYSIS

Vol. 1, No. 1, April 1971

All Rights Reserved by Academic Press, New York and London

Quadratic Functional of Brownian Motion TAKEYUKI HIDA

Mathematical

Institute, Nagoya

University, Nagoyd,

Japan

Functionals of Brownian motion can be dealt with by realizing them as functional of white noise. Specifically, for quadratic functionals of Brownian motion, such a realization is a powerful tool to investigate them. There is a oneto-one correspondence between a quadratic functional of white noise and a symmetric L 2 (i? 2 )-function which is considered as an integral kernel. By using well-known results on the integral operator we can study probabilistic properties of quadratic or certain exponential functionals of white noise. Two examples will illustrate their significance.

INTRODUCTION

T h e purpose of this note is to investigate quadratic functionals of Brownian motion. Any functional of Brownian motion may be realized by that of white noise which is, roughly speaking, the derivative of Brownian motion. We shall, therefore, start with the measure fj, of white noise. A quick review of the known results on the Hilbert space (L2) = L2(/x) and the analysis on it will be given in Section 1. Specifically, the representation of (L 2 )-functionals in terms of symmetric L 2 (i?")-functions n J> 1 is the main technique for the discussion of the present note. In Section 2, we shall deal with quadratic functionals of white noise, by using the representation by symmetric Z,2(i?2)-functions. These functions can be regarded as integral operators, and, therefore, we are able to appeal to the classical theory of integral operators. Then we shall proceed to exponential functionals whose powers are quadratic functionals (Section 3). T h e idea behind the computation is similar to the case in Section 2. Two applications of our theory will be noted in Section 4. T h e first one is the stochastic area defined by P. Levy. Formally speaking, it denotes the area enclosed by the curve of the two-dimensional Brownian motion and its chord Received November 13, 1970. AMS 1970 subject classification: Primary 60J65, 60K99; Secondary 60E05. Key words and phrases: white noise; functional of Brownian motion; modified Fredholm determinant; semiinvariant; stochastic area; equivalence of Gaussian measures.

58

59

QUADRATIC FUNCTIONALS OF BROWNIAN MOTION

within a finite time interval. Such a stochastic area can be realized as a quadratic functional of white noise. As a second application, we shall take up RadonNikodym density of a Gaussian measure equivalent to Wiener measure, where the result in Section 3 can be applied.

1. PRELIMINARIES

This section is a short exposition of some known results on white noise and square integrable functionals with respect to the measure of white noise. Let £ be a nuclear space such that

ECL\RX)CE*, where E* is the dual space for E, and let /n on E* be the measure of white noise with the characteristic functional C(£): C(f) = e x p { - 11| f ||*},

(|| |i : the L 2 (iP)-norm).

We denote by <*, | > , x e E*, £ e E, the continuous bilinear form which links E and E*. For fixed $, (x, £> is a Gaussian random variable on the probability space (E*, fi) with mean 0 and variance || $ |l2. The linear functional <x, £> extends to a random variable <x, rf) with 77 e L^R1) although it is no longer a continuous functional on L 2 (i? J ) for fixed x e E*. If {£„} is a complete orthonormal system in L2(R}), then {<x, £„>} is a system of mutually independent, standard Gaussian random variables. These (x, £m>'s span a subspace J#\ of the Hilbert space (L2) = L 2 ( £ * , ;u.). Further, finite products of the form [ ] H B l « « , $kyiV2),

Hn : Hermite polynomial

k

with Y,knk = n s P a n a subspace 3^n of (L 2 ). A member of 3/fn is called a multiple Wiener integral of degree n. T h e entire space (L2) itself admits the direct sum decomposition

(i2)=Z0/„,

^ 0 = C.

(1)

n—0

We are interested in a representation of functionals in each #Pn . In fact, there is a one-to-one correspondence between tffn and the space L\Rn) of symmetric L 2 (i?")-functions: Jtn 3 9(x) <-» i^( M l ,«,,...,«„)£L*(R")

(2)

49

HIDA

This correspondence (2) can be established by the transformation T given by (T] E

= C(i) j—JF„(Ul,

u2,..., un) |(«!) f(w2) ••• £(«„) dux du2 ••• dun .

(3)

R"

For norms of 93 and F m , we have

IMIf^aMl^llW Further detailed discussions may be found in [1, Part III]. We now briefly mention about the flow {Tt} of the Brownian motion defined on (E*, /n). T h e Tt is the adjoint of the shift St acting on E: (StMu)

= &u - t),

£eE.

For each
9(Ttx).

corresponds to

2. QUADRATIC

FUNCTIONALS

We first restrict our attention to random variables in J^ . T o each real-valued random variable
F=YlK?V»®Vn n=l

(4)

61

QUADRATIC FUNCTIONALS OF BROWNIAN MOTION

with £ A„2 < oo, where {An , ry„; n ^ 1}, AM real, is the eigensystem of the operator F such that -r\n eL^R1) with \\r)n\\ = 1 and that A„ JF(u, v) 7jn(v) dv =

-qju).

Noting that the transformation T is one-to-one, expression (4) shows that *-l),

(5)

n

where the sum in (5) should converge in J^2 . Here the eigenvalues An are arranged in order of increasing absolute value and taking multiplicities into account. As a consequence of the expansion (5), we can easily prove that the characteristic function x(%) of the random variable
= f . e x p [ i 0 ^ ) ] dp{x) = n « l + 2;*A; 1 )- 1/2 exp[ I -*A; 1 ]}, J

E

n

z: real.

(6)

If we denote the modified Fredholm determinant of F by 8(A) = S(A; F) (see, e.g., [4, Chap. VI]), the product in (6) turns out to be 8(—2iz;F)~ 1 l 2 . T h u s we have generalized the Varberg's result [5]: THEOREM 1. The characteristic function x(%) of a random variable
where F is the L\W)-function

= o(-2iz;Fyv*,

(7)

uniquely determined by (2) with n = 2.

Each
)yn = (-2r-1(n-l)lY1Mr,

n^2.

(8)

62

HIDA

Proof. Let Fin) be the w-times iterated kernel of F and let an be the trace of F . Then the following formula [4, Chap. VI] in)

7

OO

-jr log 8(A) = — Y <7n+1An, d\

| A | sufficiently small,

„ .

proves that

l°g *(*) =

—

2 l 0 g S(~~2/*)

=11,7^ n=2

= ± £ (-2)" a n (ii-l)! («)»/»!. 2 _

On the other hand, it is known that a

n = E ^t"-

Thus we have proved (8). It is interesting to observe the particular case, where F(u, v) satisfies the condition F(-u,

-v)

= -F(u,

v)

( = -F(v,

u)).

(9)

The following assertion is straightforward, and the proof is omitted. PROPOSITION 2. (i)

Suppose that F(u, v) satisfies the condition (9).

If X is an eigenvalue ofF, so is —A.

(ii) If
= 8(-2iz;

i T 1 / 2 = j n (1 + 4 * V ) ! ~ 1 / 8 .

(10)

where ± An's are the eigenvalues ofF. Furthermore, the semiinvariants have the forms

\y2m = 2*m(2m-l)\YJK2m-

(U)

63

QUADRATIC FUNCTIONALS OF BROWNIAN MOTION

Another interesting case appears in the next section, where all the negative eigenvalues of F are bounded from above by —4 : Xn > 0, or A„ < —4. We now come to a stationary stochastic process {X{t, x), — oo < t < oo} defined in such a way that X(t, x) = 9{Ttx),


(13)

where {Tj} is the flow of the Brownian motion. In order to investigate the process {X(t, x)}, we wish to know the characteristic functional Cx(g) given by Cx(£) = / . exp [i J "

X(t, x) $(t) dt] d^x),

£eE.

(14)

T o be lucky, the integral X(£, x) = / " „ X(t, x) |(f) dt is still in ^ , and, assuming thatF(u, v) eL2(Rn) corresponds to
p ^

F(u - t , v - t ) {(t) dt

(15)

—OO

corresponds to X(£, x). Thus our problem has been reduced to obtain a characteristic function of a single random variable in 3^2 , which we have discussed before.

3. EXPONENTIAL

FUNCTIONALS

Let
(16)

Obviously, f(x) belongs to (L2) and admits the transformation T defined by (3). PROPOSITION 3.

With the notations established above, we have

( r / ) ( | ) = C ( | ) S ( - 2 i ; F)~W exp £ j

-

^ fon , | ) 2 ] ,

(17)

where An's and rjn's are the eigenvalues and eigenfunctions, respectively, ofF{u, v). Proof. Since F(u, v) has the expression of the form (5), f(x) can be written in the form

/(*) = exp f-x£ A; 1 ^, Vny - 1)].

64

HIDA

We note that the 7jn's form an orthonormal system in L 2 (/? 1 ), so any £ e E may be expanded as

f = Ia»1« + r.

(18)

where £' is orthogonal to the span of the ij„'s. Thus || f ||2 = X!™ «n2 + II f II2 has to hold. With the expansion (18) we now obtain the transformation ( T / ) ( £ ) as follows:

(T/)(£) = f , exp[»'<*. £>]/(*) «W*) = (" , eX P [*Z *»<*> ^n) - *Z ^ K * . ^«>2 - 1)1 J

J

l

E

n

n

• exp[z<x, £'>] dp(x). Since (x, ij„>'s and (x, f'> are mutually independent, the last integral is equal to I

exp[«<*, f >] ;„> - i\~\x,

= C(f) n | ^ 7 = J "

2

Vn)

+ /A"1] d/j.(x)

exptffl,,* - (*V + |)f 2 ] ^ • exp^A; 1 ]}

= C(i') EI j(l + 2iKT1/2

expfA; 1 ] • exp [ - \ y ^ p ^ p r ] j

= C(?) S(-2i; FT1*

exp [ £ y ^ g ^ r ] • exp [ - \ £

= C(£) 8 ( - 2 * ; F)~^

exp [ £ y ^ J ^ r ] >

< ]

which was to be proved. We define a transformation of F to F in the following manner: If F = (x) rj„ , then F is given by

F = I i^-i^®^-

(19)

F is no longer real, but it is a symmetric L 2 -kernel which shares the eigenfunc-

65

QUADRATIC FUNCTIONALS OF BROWNIAN MOTION

tions rjn with F. As an integral operator, F may be characterized by the following relations: (i) range.

As integral operators, F and F are commutative and have the same (20)

(ii) F(I + 2iF) = iF. Now we have the main theorem. THEOREM 2. Let f(x) = exp[i
( r / ) ( 0 = C ( 0 8 ( - 2 » ; F)-V2 e X p [ [ J F ( u , v) f(«) « » )
/ 2 B + 1 (x) = 0,

»>0,

and (r/UXfl = C(fl 8 ( - 2 i ; F ) " 1 ^ " ! ) - ^ ® . f n ® W » ) ,

» > 1-

Remark. T h e superscript «(x) stands for the w-times tensor product. By the formula for (T/ 2M ) we can easily see that the L 2 (/? 2n )-function corresponding t o / 2 „ is the symmetrization of S(—2t; F)-li\nV)-1Fn®. Proof of Theorem 2. T h e asserson (i) is a rephrase of Proposition 3 by the use of the notation given by (19). For the proof of (ii), we expand the exponential part of (rf)(i) like a power series expansion: ( r / ) ( 0 = C(0 S ( - 2 i ; F)~^

£

[jjP(u,

v) £(«) f(e)
Each term is a homogeneous functional of f. The term of degree 2n has to be (T/ 2 „)(£) (for detailed discussion, see [1, Section 6]). T h e rest of the proof is now obvious. We then consider the case where g(x) = exp[

(21)

is in (L2). It is easy to prove that a functional of the form (21) is square integrable if F(u, v) corresponding to
66

HIDA

Then g(x) belongs to (L2), and we have faXfl

=

C

(^) 8 ( - 2 ; F ) " 1 / 2

ex

P [//

G u

('

v

) W

&v) du dv\,

(22)

ivhere 0 is a real L2(R2)-function given by G(u, v) = £ (A. + 2)-lVn(u),,(«).

(23)

n

T h e proof is, formally, quite similar to that of Proposition 3. A crucial difference is the fact that we always have to be careful about the integrability of the integrand and the convergence of the product. As a consequence of this result, we obtain the projection gn(x) of g(x) on 3^n in terms of (rg„)((). In detail, g0(x) = 8 ( - 2 ; F)-W,

g2n+1(x) = 0,

n > 0, (24)

(rg2n)(0 = C ( 0 S ( - 2 ; F ) - V > ! ) - i ( ( > ® ?»®)LHR^

4.

4.1.

.

APPLICATIONS

Stochastic Area Due to P. Levy

With Levy's notation we first give an intuitive meaning of the stochastic area defined by the standard two-dimensional Brownian motion. Let (X(t), Y(t)), t J> 0, be the ordinary plane Brownian motion. Similar to a smooth plane curve, P. Levy has defined S(t) by

S(t) = if

(X(s) dY(s) -

Y(s) dX{s)).

(25)

(see, e.g., [3]). The integral can not be defined for an individual sample curve, but it is defined as a stochastic integral. We may, therefore, consider S(t) as a stochastically defined area enclosed by a Brownian curve up to moment t and its chord. In our set-up, S(t) can be realized as a quadratic functional on the space E*. Indeed, we realize X(t) and Y(t) by <*, X[o,*]> a n d <X X[-*.o])> respectively. Thus the integral S(t) based on them is now a functional, call it tp{t, x), in 34?2 . For simplicity, we set t = 1. It is easy to see that the following F(u, v) eL2(R2) corresponds to
67

QUADRATIC FUNCTIONALS OF BROWNIAN MOTION

F(u,

v)

FIGURE

'i

F{u, v) =

I 4 '

;o,

1

if uv ^J 0, u, v ^ 1, if ww < 0, tt, w ^ — 1, elsewhere.

—V < M,

—» > u,

(26)

Now we prove THEOREM 3. The characteristic function semiinvariants are given by

of
E(exp[izcp(l, x)]) = {cosh(af/2)}_1, jy2n+i = 0,

(27) (28)

iy2n = (2^-l)fl„/(4«), where the B's

and its

are the Bernoulli numbers.

Proof. T h e kernel F given by (26) satisfies the condition (9), and it has eigenvalues 2(2« — \)TT, n = ± 1 , ± 2 , . . . , the multiplicities of which are all 2. By using the formula (10) for the characteristic function, we have -1/2

X(«) =

E[ (1 + <M{4(2« -

1)V})*

= l/cosh(*/2).

^ n

T h e formula (11) easily leads us to the result (28). Remark. T h e stochastic process {S(t), t ^ 0} can also be discussed. We note that theL 2 (.R 2 )-function.F(£; u, v) corresponding to 0} of kernels gives us full information about {S(t); t > 0}. For example, dS(t) in terms of F(t; u, v) illustrates interesting properties of the stochastic area, although the expression is intuitive.

68

HIDA

4.2. Gaussian Measure Equivalent to Wiener Measure Let X(t), 0 < t ^ 1, be the standard Brownian, and consider a Gaussian process Y(t), 0 < t < 1, given by Y(t) = X{t) -

f j fF^s,

U) dX{u)\ ds,

(29)

where i ^ is an L 2 -kernel of Volterra type (i.e., Fx(s, u) = 0 if u > .?). It is well known that Y{t) defines a probability measure on C([0, 1]) equivalent to Wiener measure and the density can be expressed in the form exp jJ"' j'F^s,

u) dX(u) dX{s) - | j

1

(J'F^S,

U) dX(u)^

ds\

(30)

(cf.,[2]). We are now ready to discuss the functional (30) as an application of Section 3. As before, we can realize X(t) and Y{t) as functionals on E*. In fact, we may set X(t) = <*, X[0>(]> and Y(t) = <*, X[0^} — J"Q ^F^S, •)> is, respectively. Define F(u, v) ( e Z % 2 ) ) by v

'

\FAu, v), (Fx(», w),

if if

0 < v < w, s > 0 0.

Then, the density (30) has an expression of the form g(x) = exp[e/.(x)], where ifi0 = ^(JQF^S, 2

4>(x) = 0 + ip2(x), and ^i2(x) is

uf du) ds = \\F1\\^2{R2)

(31) an

^-functional

F(u, s)F(s, v) ds\ '

(32)

2

to which the L (^ )-kernel G(u, v) = - I \F(U, V) ^ '

f

J

«v«

corresponds (u \f v means max{w, v}). T o analize the density g(x), it suffices to discuss the kernel G(u, v) given by (32). We note that F itself may be expressed as the sum F = Fx + Fx* Since ^]iynF(u, s)F{s, v) ds = J~ F^u, kernel G as follows:

(* denotes the adjoint). s)F1(s, v) ds holds, we can express the

G = -UF1+F1*-F1*FJ.

(33)

69

QUADRATIC FUNCTIONALS OF BROWNIAN MOTION

Each kernel arising in this equation is of Hilbert-Schmidt type. As in Hitsuda [2, Section 5], we see that (/ — Fi*)(I — -^I) is strictly positive definite. Therefore, it is proved that the eigenvalues of the kernel G are either positive or bounded above by —2. Now we conclude the following result: If the kernel G(u, v) given by (32) has no eigenvalue A such that —4 ^ A < —2, then the density g(x) belongs to (L2) and the projections of g(x) on £Fn are obtained by the formulas (24) with the kernel 0 given by (23), where the i7„'s should be replaced by the eigenfunctions of G. Such a consideration seems to be useful for the integration of functionals of a Gaussian process Y(t) equivalent to the standard Brownian motion.

REFERENCES

[1] HIDA, T. (1970). Stationary Stochastic Processes. Princeton Univ. Press, Princeton, N.J. [2] HITSUDA, M. (1968). Representation of Gaussian processes equivalent to Wiener process. Osaka J. Math. 5 299-312. [3] LEVY, P. (1954). Le Mouvement Brownien. Gauthier-Villars, Paris. [4] SMITHIES, F. (1958). Integral Equations. Cambridge Univ. Press, Cambridge. [5] VARBERG, D. E. (1966). Convergence of quadratic forms in independent random variables. Ann. Math. Statist. 37 567-576.

59

Generalized Brownian Functionals* TAKEYUKI HIDA

Department of Mathematics, Faculty of Sciences, Nagoya University, Nagoya, Japan and L. STREIT

Fakultat fur Physik, Universitat Bielefeld, D-4800 Bielefeld 1, Germany If one tries to visualize the Feynman integral [1] as an average over fluctuating paths y(t) such as e.g. y(t) = y0(t) + aB(t) (1) with Brownian motion B(t) = JQ B(t)dr, B Gaussian White Noise, one is naturally led to consider nonlinear functionals of Brownian motion, such as e.g. the exponential of the action i/n J L(y,y)dr or J-function pinnings S(y(t) -

x).

A large class of nonlinear Brownian functionals is provided by the Hilbert space % of finite variance functionals [2] U = L2(dfi),

(2)

where fi is the White Noise Measure characterized by C ( / ) = /V<^>d/z[5] = e"5 H/H2.

(3)

However, it is not large enough to accommodate the above examples. This follows from a study of the source functionals given for any $ S % by

r$[j] =

E{e^'j)§)

=

2 n e-h\m j2i

„

oo

„=0

jRn

n

Fn(t1,...,tn)l[j(tv)dtv.

(4)

v=\

Theorem (Hida [3]): I. r : y. -f^ TZ is an isomorphism between the White Noise Hilbert space % and the Hilbert space TZ with reproducing kernel

K(j1,j2)

=

C(j1-j2).

•First published in Lecture Notes in Physics, No. 153 (1982) 285-287

60

I I . An isomorphism a between H and Fock space T oo

a : H <-» T = 0

v^!i2(i?n),

n=0

is induced by (4): a : $ o (F„)g° . Hence, to discuss generalized (and smooth) Brownian functionals it is useful to embed Fock space in a nuclear rigging [4] To C T C Tl which through the isomorphisms a and T produces nuclear riggings of the RKHS 1Z and of H ILQ

C

T~L

C riQ .

An example is provided by the "minimal rigging" of [5]. It is not hard to verify that the (normal ordered) square of White Noise, - formally / : B2(T) : dr = / B 2 ( r ) d T - E(\-) its normalized exponential - formally Afexp ( a / 5 2 ( r ) d r ) = eaJ'&^dT/£(.(.) and the "pinning" S(B(t) — x) can be realized as generalized Brownian functionals in this sense. The space of smooth functionals (To resp Ho) is, as usual, a matter of choice and convenience, see e.g. [6]. In particular riggings somewhat "larger" than the minimal one can be found which will include exponentials such as e1^3'^ for smooth j as a test functional, without losing the above examples from the corresponding space of generalized functionals. A prominent example in this framework is the "Feynman integrand" / = Af exp ( \ j \

my2dr + \ J

B2(r)dr\

S(y(t) -

y2).

1 In

With the trajectories y(r) = yo(r) + (•—) B(T) starting from y(0) = j/o(0) = y\ one finds the expressions E(I) = G0(yi,y2;t) and E(Ie-UVdT)

=

G(yi,y2;t)

for the propagator of a free particle and for that of a particle in the field of a potential V; as usual V is chosen as the Fourier transform of a finite measure so that Dyson series techniques are available [1,7]; for the harmonic oscillator potential E can be evaluated in closed form [7], The necessity of the second term in the exponent of I is noteworthy. Effectively it cancels the exponential fall-off of the White Noise measure /x, mimicking in this way the formal infinite dimensional "flat" measure Uo
61 paths of the form (1) with which one would recover the quantum mechanical Green's function in the limit of infinite fluctuation strength a. In either case, the sure part yo(r) need not be chosen as either a free or a classical trajectory. In fact, it is quite arbitrary. This reflects the translation invariance of the Feynman "measure". The classical path, plus an expansion of V(y(r)) around yd to second order can be evaluated in closed form and yields the WKB expansion [7]References 1. S. ALBEVERIO, R. H O E G H - K R O H N : Mathematical Theory of Feynman Path Integrals, Lecture Notes in Mathematics, vol. 523, Springer 1976. 2. T. HlDA: Brownian Motion. Applications of Mathematics, vol. 11, Springer 1980. 3. T. HlDA: Casual Analysis in Terms of White Noise, in "Quantum Fields Algebras, Processes", (L. Streit, ed.), Springer 1980. 4. N.N. BOGOLUBOV, A. A. LOGUNOV, I.T. TODOROV: Introduction to Axiomatic Quantum Field Theory, Benjamin 1975. 5. P . KRISTENSEN, L. M E J L B O , E.T. POULSEN: Commun. Math. Phys. 1, 175 (1965). 6. I. K U B O , S. TAKENAKA: Proc. Japan Acad. 65A, 376, 411 (1980). 7. L. S T R E I T , T . HIDA: Generalized Brownian Functionals and the Feynman Integral, Bielefeld preprint BI-TP 81/26.

Exponential

functions and Brownian functionals

569

THE ROLE OF EXPONENTIAL FUNCTIONS IN THE ANALYSIS OF GENERALIZED BROWNIAN FUNCTIONALS MIDA

T.

0. Introduction. Nonlinear functionals of a Brownian motion {B (t), I s R1} may d be expressed as those of the white noise {B (t), t e R1}, B (t) — —TT- B (t). Further, with the assumption of finite variance, they can actually be realized as members of (L2) = = Ll ((?*, p,), where S* is the dual space of the Schwartz space
(2)

(L*) = 2 e 9en,

•where &£n is the space of the multiple Wiener integrals of degree n. The integral representation of a Brownian functional, i. e. a member of (L z ), is given by (<^q>(x)dli(x), dp*

56#,


(3)

To be more precise, if q> is in ffln, then we have f (*"
<4>

F (5) = J • • • J ^ ("i. ••••"„)& («i) • • '5 K><*""• where F is a member of L a (ii71) = {f e= i 2 (i?71); f symmetric} and || to ||(LS) = = V"ni||/ , || „ . The function F is said to be the kernel of the integral representation, and the functional U is called the V-functional of (p. The {/-functional of a general 9 with

(5)

where 3K^ is the class of .^-functionals whose kernels are in the symmetric Sobolev space H{n+1)l2 (Rn) = fl-/2 (i}«) r\ 1? (Rn) and is topologized so as to be isomorphic 00

to H<-n+t'>12 (Rn), and where ^Sf(-») is the dual space of #?(»>. Set (1,2)+ = JJ ffi ^ L " ' n

n

and

n=*0

introduce (Ll)~, call it the space of generalized Brownian functionals, as the dual space of (La)+. A member of (i 2 )+ naturally plays the role of a test function. The idea behind such a construction of (L 2 ) - may be found in [2] and [3]. Typical examples of a generalized Brownian functional are polynomials in B (t)'s that are realized in .??£"">. Let Hn (a; a2) be the Hermite polynomial in a of degree n with parameter a 2 . Then we see that Hn (AJS/A; 1/A) belongs to #en. Letting A -•> {t}, the limit, which is formally written as Hn (B (t); ildt), no more remains within #en, but is found in 3%£"H). Since each x in S* with ft is viewed as a sample function of B (t), the polynomial in B (t) given above may be written as Hn (x (t); ildt), although x evaluated at t has no exact meaning. For details, see [2, Chapt. 8]. After polynomials naturally come exponential functions of the B(t)'s, which are of course generalized Brownian functionals. We should, in this note, like to emphasize the importance of the role of exponential functions in the so-called causal calculus of generalized Brownian functionals, where the development of the time is involved explicitly. We therefore start with a quick review of exponential functions, then proceed to some basic properties of them in p. 1, and then we shall discuss differential calculus applied to exponential functions in terms of {B (t)} in p . 2. Some concluding remarks will be given in the last section.

570

Hida

T.

1. Exponential functions. A Brownian functional of the form const exp [a <£, |> ], a e C, g s S,

(6)

is called an exponential function on <$>*. As a random variable it extends to a functional with % in L* (i?1). For further generalization we are now able to introduce exponential functions of B(t)'s themselves without smearing them. A simplest example seems to be exp, [iXB (t)] (or exp [iXx (t)], x ^ S*, X e R1, however it is not well defined unfortunately. An approximation to this function leads us to form a renormalization of the formal expression exp [iXB (/)]. Take an ordinary exponential function exp [iX&B/A] and apply the transformation 3~ to obtain exp[-V 2 ||Xx A /A + !jf] = exp [ - ^

- X (x A , 5 ) / A ] C (%),

XA is the indicator function of A. We now expect that the limit of exp [iXABI'A + + k2/2A] would be a generalized function with the ^-transform expt—X\ (()] C (|). In fact, this is true. The generalized Brownian functional thus obtaintd is expressed in the form tp^ (t) = exp [iXB (t) + l?,2dt\ (7) (or :exp [iXB (t)\: in physicists' notation) and
V (|) = exp HI (t)].

(8)

It is noted that the generating function of the Hermite polynomials Hn(B (t); ildt) introduced in p. 0, is nothing but the generalized^unctional (p^ (t) given by (7): 00


(9)

We then come to a slight generalization of (7):

*»(«!.•••.»») = e*P [ ' S h B (';)+ S ^ 2 * / ] '


i

t'-s different, The ^"-transform of (f^

<10>

i

X, e R1.

^ (tlt • . ., tn) is given by exp \ — 2 ^ j 5 ('j)l C (£) and

the corresponding [/-functional is therefore

V (l)= exp [ < 2 ^ ( 9 ] •

(H)

We know that'the algebra over C generated by exponential functions of the form (6) is dense in (£2) (e. g. see [2, Chapt. 4]). Similarly in spirit, we have proved folloving result. Theorem. The collection G> = {(f^ ^ (tu . . ., tn)\ t'^s different, tj e R1, Xj e R1} 2 is dense in (L )~ in the sense that a test functional which is orthogonal to any member in G> must be identically zero. P r o o f . W e m a y a s s u m e t h a t a test f u n c t i o n a l i|> is an ^?^*'-functional for s o m e n w i t h a k e r n e l F (ult

. . ., u„) <s #("+D/2 ( R « ) . B y a s s u m p t i o n we h a v e «q)^

^

(tlt

...

. • •> tn), ^ » = 0 for a n y choice of different us and ^ s , w h e r e « , » is t h e c a n o n i c a l b i l i n e a r form c o n n e c t i n g ( L 2 ) + and ( £ 2 ) - . T h i s m e a n s t h a t m ni+...+n^=n

n2

"J

j=i

for any choice of Xf. Therefore, we have F {u\, . . ., un) = 0, which was to be proved. With this important properties in mind, we introduce a class of generalized random measurs that come from the Taylor expansion (the product of the expansions like (9)) of % , . . . , * , „ Ci. • • •' '")• Set Mm (dt) = m\Hm (B (t); ildt) dt = m\Hm (x (t); ildt)dt,

x e #*,

Exponential

functions and Brownian

junctionals

571

and set I I MM„ (dt.\. ^T-T -Mn {*,) = I n nMt})' ,=1

v

t',s are are different, different. *>

0

(12)

otherwise,

Then f [ • Af n .(^.) plays the role of a random measure (see [2, Chapt. 8], and [41), and is called a generalized random measure. After P. Levy [1], an ^?^""^-functional which is expressed as an integral with respect to a generalized random measure given by (8) with 2 " i = » is called a normal functional of degree n. A sum of normal functionals of various }

degrees is simply called a normal functional. E x a m p l e of a normal functional of degree re: 9 = l--i\G(uu...,uk)Mni(da1).....Mnk{duk),'2in)

= n.

(13)

The class of normal functionals of degree n never covers the entire space 3f£^~n\, however the class occupies the important part from the viewpoint of the causal calculus. Returning to the functional q^ x (tlt . . ., tn), we may think of an analogous expression exp p ^ j ^ j A j ( A j 5 / A j) + S ^ i A j ' 2 / 2 A j l

a n deven t n e limit

t t cp (t) = exp [i jj / (u) B (u) du + i/ a {j / («)2 du] , o o

(14)

where ^j^j^A- approximates / (u) over the interval [0, t\. As is easily seen, 9 (t) is j

]

a martingale relative to- the increasing o-fields generated by {B (u), u < (}, and our observation says that the martingale tp (*) is viewed as a continuous product of exponential functions involving all the variables B («), u < t. 2. Differential calculus. Since {B (*)} is taken to be the system of variables of generalized Brownian functionals, we shall be able to introduce the differential operator dldB (i). D e f i n i t i o n . Let 9 be an (L 2 )"-functional and let U (|) be the corresponding {/-functional. If U (£) has the functional derivative UL (t) in the Frechet sense, and if here exists an (L2)~-functional cp' whose {/-functional is U\ (t), then 9 is a said to be differentiable at t with respect to B (t). The 9 ' is denoted by 9 . We often denote dB(t) the operator simply by 3(. We observe a simple example. Set

+ 4 - 1 / 1 1 ' ] '

/sL2(fli),

(15)

Then the {/-functional of 9 is U (I) = exp [i (f, g)].

(16)

Obviously, ui (t) exists and is expressed in the form U't (t) = if (t) U (I).

(17)

This means that 9 is differentiable at t with respect to B (t) and 9(9 = if (t) 9.

(18)

As for the mean of the random variable we have E9 = 1.

(19)

The technique developed in [4] tells us that the equations (18) and (19) characterize the exponential function 9 given by (15). In fact, let {Fn; r\ J> 0} be the sequence of kernels determining the V (%):

572

Hida T.

Then we must have CO

U'%(t) = yA n $ . . . ^Fn ( » ! , . . . , un_v t) 5 ( U l ).. .1 (un_J

du1"1.

If we assume (17) and (19), we immediately obtain the formula (16). We are now ready to discuss our exponential functions of the form (10) by using the differential operator dt. Since the (7-functional is of the form (11), the functional derivative V'. (f) only exists as a generalized function of t for every | . In fact, it is of the form V'iif)=l^fi(t-tj)U(l), i where 6 is the 6-function. Namely, we have

»,«p».

xjh

(20)

'n) = ' 2 ¥ ( ' - 9 , P x .

»»

ft-••••'»)•

<21>

i As in the example discussed above, we can easily prove the following proposition. Proposition. / / a generalized Brownian junctional (p satisfies the equation i then, under additional condition that <<> = 1, tp is given by the formula (10). The equation like (21) involving the operator dt is often called a stochastic partial differential equation. Being inspired by the remark mentioned at the end of the last section, we are in search of a stochastic partial differential equation that characterizes the martingale defined by (14). To avoid complication we consider a martingale t

<{>(«) = e x p \i%\ B (u) du + ' - j H , o It satisfies the equations (23), (24); W W = l

*>0.J

0, otherwise, |Eq>(t) = l .

(22)

(23)

(24)

These equations actually characterize the martingale cp ((). Now set

(25)

It is interesting to see some similarity between the ordinary trigonometric functions and <J>c> 9s> when dj is viewed as the ordinary differential operator. Namely, . a tt < P. = X
0
3. Concluding remarks. We have so far observed some important properties of exponential functions, both ordinary and generalized ones, and discussed their characterizations in terms of stochastic partial differential equations. Similar approach has been made in [4] by the author, where the use of the equations has first been proposed, and he believes that this is one of the powerful tools of our analysis. Another interesting and systematic approach to the causal calculus has been established by I. Kubo and S. Takenaka [5]. The author has hopes that these results would serve the purpose to develop the causal calculus of generalized Brownian functionals, where the passage of time is always taken into account explicitly. REFERENCES 1. Levy P. Probleme concrets d'analyse fonctionnelle. Paris: Gauthier —Villars, 1951, 2. Hida T. Brownian motion. Transl. from Japanese. (R. Appl. Math., B. 11.) Berlin etc.: Springer — Verlag, 1981. 3. Hida T. Analysis of Brownian functionals. 2nd ed. (Ser. Carleton Math. Lect. Notes, J6 13.) Ottawa: Carleton Univ., 1978, 83 p.

66

Large deviation probabilities for U-statisticj

573

4. Hida T. Causal calculus of Brownian functionals and its applications.— In: Proceedings of the International simposium on statistics and related topics. Ottawa: 1980. 5. Kubo I., Takenaka S. Calculus on Gaussian white noise. I; II.— Proc. Japan Acad., 1980, v. 56, ser. A, JV» 8, p . 376—380; JVs 9, p . 411—416. nocTynnna B pen;aKn;HK> 10.III.1981

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T. (HArOX,

fjnOHHH)

(PesmMe) PaccMaTpHBaeTCH coBOKynHOCTb aucnoHeHT OT JIHHCHHBIX (jiyHKHHOHajioB OT TpaeitTOpnfi BHHepoBCKoro nponecca, o6pa3yiomaH njioTHoe nosMHOJKecTBO B npocTpaHCTBe L s o6o6meHHbix (jiymmnoHajioB OT BHHepoBCKoro nponecca. flna 3KcnoHeHn,HajibHbix <j>yHKn;HH BbiBOBHTCH AHi|"i>epeHHHajibHBie ypaBHeHHH, HBJisnomHecH TexHHiecKHM annapaTOM B HCHncjieHHH HejiHHeflHiix <j>yHKn;HH OT BHHepoBCKoro nponecca.

PREDICTION THEORY AND HARMONIC ANALYSIS The Pesi Masani Volume V. Mandrekar and H. Salehi (eds.) © North-Holland Publishing Company, 1983

CAUSAL CALCULUS AND AN APPLICATION TO PREDICTION THEORY T. Hida Department of Mathematics Faculty of Science Nagoya University Chikusa-ku, Nagoya 464 JAPAN Some developments of the causal calculus of Brownian functionals are discussed by using the d i f f e r e n t i a l operators 3*'s

d. = 3/3B(t), t € R , and t h e i r adjoints

as well as polynomials in them.

As an application

we can solve the equation ±

X(t) = -xX(t) + f ( 3 * ) ( b X ( t ) + b ' )

and discuss the prediction problem for the solution process §0.

{X(t)}.

Introduction We shall discuss the calculus of functionals of Brownian motion, call

them Brownian functionals, in particular the so-called causal calculus, where the time development is always taken into account explicitly.

We shall also

discuss an application of the calculus to the prediction theory for some stochastic processes. In line with the causal calculus, the author has proposed Stochastic Partial Differential Equations by the use of the differential operators like 3/3B(t), where Brownian motion

B(t) = -rr B(t) {B(t)}

is the white noise coming from a standard

(see T. Hida [3]). The equations not only describe

random phenomena depending on

t, but also characterize some classes of

Brownian functionals as well as generalized functionals.

Other interesting

developments of the causal calculus made so far are found for instance in the applications to quantum mechanics (see [4], [9]). The main idea behind the causal calculus is to take

(B(t)}

to oe the

AMS 1970 Subject classification, Primary 60H99, 60G25. Key words and phrases, Brownian functionals, stochastic partial differential equations, causal calculus.

T. Hida

124

system of variables of Brownian functionals.

By doing so, the time shift is

represented by white noise in such a way that to

t+h.

B(t) -* B(t+h)

as

t

proceeds

From this idea, it seems quite natural to think of that the causal

calculus would be one of the powerful tools in the prediction theory.

In fact,

we shall be able to show in section 3 that the calculus serves as a useful method in prediction theory for some classes of generalized stochastic processes. We expect that similar, in spirit at least, technique would be established in more general approach to the nonlinear prediction theory. Before we come to this main topic, background of the causal calculus together with some remarks will be summarized in sections 1 and 2. §1.

Background Let

(B(t), t 6 R } be a standard Brownian motion and take the time

derivative

B(t) = -nr B(t)

to have the white noise

to analyze functionals of the form the probability distribution

y

of the white noise.

generalized stochastic process, y functions on

R , say by

S*

(B(t), t e R }.

In order

cp(B) =
{B(t)}

is a

is supported by a space of generalized

the dual of the Schwartz space

5.

In fact,

starting with the characteristic functional C(e) E E(exp[i B U ) ] ) = exp[- ^|5|| 2 ], II II the L^R 1 )-norm,

(1)

the measure

y

is determined in such a way that

(2)

C(e) = [ exp[i<x,5>Jdy(x). J 5* 2 As soon as

A member of

(L )

y

is given, we form the Hilbert space

2

L (5*, y) = (L ).

is now viewed as a realization of a Brownian functional

with finite variance, because with

y, almost all element of S*

is thought of

as a sample function of B(t). 2 The space (L ) admits the Wiener-Ito decomposition: (3) (L2) = I • H , n=0 where n.

H

n

is the subspace consisting of multiple Wiener integrals of degree

Important tool for our analysis is the

(4) If T (5) where

T-transform given by

(Tcp)U) = f e i<x,C> cp(x)dp(x), cpe ( L 2 ) , e € 5. is restricted to

H , we have {TV)U)

= i n C(?)U(5),

9 6 Hn,

Causal Calculus and an Application to Prediction Theory .U(e) = [ ••• [ F ( u r - " , u ) e ( U l ) - - - 5 ( u )du n ,

(6

'

R1 F € l / ( R n ) (= the space of symmetric

The correspondence between ||cpl|

125

cp and

L 2 (R n )-functions).

F is one-to-one, and further holds

„

= i^nT||F|| ~ . (Integral representation. See, e.g. [ 2 ] . ) (I) I (Rn) f u n c t i o n a l U i s o f t e n c a l l e d the U-functional associated w i t h cp.

The

The space o f U - f u n c t i o n a l s associated w i t h H - f u n c t i o n a l s i s denoted by F , and i s topologized so as t o hold <7>

F

For a general

n=Hn

•fiiiV).

cp=£ cp , cp € H , we associate

given by (5) for

cp .

UU) = I ^ ( c ) ,

An alternative way of defining

wnere

U for

U

ls

cp is due to

the 5-transform introduced by Kubo-Takenaka [ 5 ] : (8)

(&p)U) = CU) [

}s*

( A l s o , see Kuo [ 6 ] ) .

§2.

Causal calculus The basic idea of the causal calculus was to take {B(t)} to be the system of variables of Brownian functionals, so that we have naturally been led to introduce the class of generalized Brownian functionals (see [2, Chapt. 8] or [1] for details). The following idagram tells how to introduce the space of generalized Brownian functionals of degree n. /nT f f ^ / 2 ( R n ) ~ H<-">~ F < - n ) /nT L 2 (R n )

^J

vVH(n+1)/2(Rn)

~ H

~ F

(n =~ Hn > ~ = Fn^ ,

where H m (R n ) = H (Rn) n L 2 (R n ), m > 0, H (Rn) being the Sobolev space on Rn of order m, and where H (R ) is the dual of H m (R n ). The space of test functionals is taken to be

(9)

(L 2 ) + = I • f/n) n 2 +

and the class of generalized functionals is the dual of (L ) by (L 2 )-.

and is denoted

T. Hida

126 Remark.

More general class of Brownian functionals can be defined by {vn''s

introducing a stronger topology to have the weighted direct sum of the in place of (9).

(See Streit-Hida [ 9 ] ) .

The ddifferen i f f e r e n t i a l operator

3. = 1

S-transform.

is defined by using the 3B(t)

Namely

V

3B(t)

provided that there exists the Freche't derivative associated with

cp.

cp 6 (L 2 )~ U'(t)

of the U-functional

This operator is in fact a p a r t i a l d i f f e r e n t i a l operator 2 + (L ) . The adjoint operator, denoted by

and i t s domain certainly incudes 3*

or

? + UT

(

)*, is defined via the canonical b i l i n e a r form

< , >

connecting

a6

^K and (L2) :

(10)

cp € ( L 2 ) + , i> 6 ( L 2 ) ~ .

O t q>,i|i> = ,

It is proved that creation operator.

3, is an annihilation operator, while

3* is a

This can be clarified in terms of the kernel of the

integral representation.

Namely, if cp is in fu

n

®^x)

and has

kerne

^

F e ^ ^ V ) , then n(6 t * F)(u 1 ,--- > uf)_1) = nF(u 1 ,---, uf)_1, t) and 6. » F ( u , , are associated with

3fcp

, u

and

, ) = the symmetrization of

3*cp, respectively.

Thus we have the canonical commutation relations between (11)

M F

[3 , 3*] = 6. I , s t t ,s

I

3.

and

the i d e n t i t y .

As a result, if f and g are polynomial, then (12)

[3 S , f(3*)] = « t ; S f'(3t).

(12)' where

[g(3 s ), f

^=«t,s9'(s).

and g' are the derivatives of f and g, respectively.

Multiplication by B(t) can also be defined: (13)

B(t)cp = (3 t + 3*)cp.

A plausible interpretation may be given i f we observe the case where a Fourier-Hermite polynomial.

cp is

3*

Causal Calculus and an Application to Prediction Theory

1

We then come to the conditional expectation of generalized Brownian functionals r e l a t i v e to the family of the a - f i e l d s 1 c ( - » . t ] } , t 6 R . To t h i s end, we f i r s t give Definition 1.

8 (B) = a { B U ) ; supp 5

A generalized Brownian functional

B t (B)-measurable, i f for every

9

is said to be

n

= 0 holds whenever the kernels F for ty 6 H is supported by the complement of the set (-•», t ] n . Definition 2. The conditional expectation of a generalized Brownian functional

cp r e l a t i v e to

8 t (B)

is a functional

cp(t)

s a t i s f y i n g the

conditions i)

cp(t)

ii)

is

8. (B)-measurable

=

holds for any

We prefer the notation

E(cp/8.)

8 t (B)-measurable test functional

to express the conditional expectation,

as in the case of ordinary Brownian f u n c t i o n a l , although

cp is a generalized

one, and i t well serves in prediction theory as we shall discuss in the next section. §3.

Application to prediction theory We s t a r t with the following formal stochastic d i f f e r e n t i a l equation.

(14)

where

^

f

X(t) = -AX(t) + f ( B ( t ) ) ( b X ( t )

is a polynomial.

+ b'),

A > 0,

This equation describes a random evolutionary

phenomena with the inference expressed by a polynomial in white noise. particular (15)

If in

f(B(t)) = B(t), then the equation may be understood as dX(t) =-XX(t)dt + dB(t)(bX(t) + b ' ) ,

dt > 0,

and we are given a stationary stochastic process as a solution to (15). For a higher degree polynomial

f such an understanding as (15) is impossible.

However, with the spirit as in (15), the equation (14) may be paraphrased rigorously in terms of the causal calculus.

Namely we propose a stochatic

partial differential equation dTX(t)

(16)

=

-AX(t)

+ f

(3£)(bX(t) + b')

(cf. [3]). It is our main purpose to show how to obtain the stationary solution to the equation (16) in our setup.

It should be noted in advance that the

128

T. Hida

solution is no more an ordinary process, but i t lives in

2_ (L ) .

In terms of U-functional, the equation (16) is expressed in the form

(17) where

^ut(g) U t (s)

equation for

a

- x u t ( e ) + f(e(t))(but(5) + b1),

is the U-functional associated with t > 0

X(t).

e e s,

The solution to t h i s

with the i n i t i a l data U 0 U) = c,

is given by ( i n a similar manner to [ 5 ] , I I I ) (18)

u\(5) = b'exp[-xt+b[ 1

J

f ( c ( u ) ) d u ] - { ^ - +f J

o

f ( 5 ( u ) ) • expCXu-bf f U(v))dv]du}, J

o

o

t > 0. The stationary solution may be given by (19)

U t U ) = b'exp[-xt+b| f U(u))du]-[ f(s(u))exp[xu-b[ fU(v))dv]du.

With these observations the stationary generalized stochastic process satisfying the equation (16) can expressed in the form (20)

X(t) = I n=l

where

Mf(du )

e"Atf

^X—

J

"•

-<»

•••[ expCAmin J

-»

l<j
u,]M (du, ) • - . , . . M f (duJ J

T

T

'

n

is the generalized random measure (see [ 2 ] , Chapt. 8) given

by the following formula. N

M f (du) =

I

, 1 a k k:H k (B(u) ; ^ d u , i f

N

f(x) -

I

k a,/.

Being inspired by P. Masani and N. Wiener [8], we are now ready to discuss the predictor for X(t). It is known that, in a particular case f(x) = x, the nonlinear predictor coincides with the linear predictor and the predictor can easily be obtained. For the process X(t) given by (20) we have the explicit form of the predictor. It is given by the conditional expectation E(X(t)/B s (M f )),

t > s,

where B (Mf) is defined in a similar manner to the case of B (B) since Mf(du) is a sum of generalized random measure. The actual form of the predictor is of the form (21)

E(X(t)/B (Mf)) - I ^ r — e " U [ •••[ exp[xmin u ] M f ( d U l ) n=l ' J-co J-o> l<j
M f (du p ).

Causal Calculus and an Application to Prediction Theory Concluding remarks, i )

1

Once the formal expression (14) is expressed

in the form (16) in terms of the causal calculus, we are ready to generalize the stochastic d i f f e r e n t i a l equation (14). -— X(t)

in (16) with

For instance, by replacing

l_ t X(t), where

H • J, -^>"-k with the requirement that a l l the roots of

N-k I a, A = 0

Another generalization is towards the case where and

X(t)

are r e a l , negative. is Hilbert-space-valued

A is a positive operator. ii)

An interesting observation i s a s follows.

Apply the operator

3

to (16) to have ^Ys(t) where

Y (t) = 3 $ X(t).

= -AY s (t) + f ( 3 * ) b Y s ( t ) + 5 t i S f ' ( 3 * ) ( b X ( t ) + b ' ) , If we set Z $ (t) = e A t Y s (t)

and if t > s, then we

have ^ Z s ( t ) = f(3*)bZ s (t),

t > s.

This is a slight generalization of the example of a stochastic equation discussed in [2] Chapt. 4, and we obtain a generalization of a martingale. By using this result we can present an alternative method of solving the equation (16). References 1.

T. Hida, (1978), Analysis of Brownian functionals.

Carleton Mathematical

Lecture Notes no. 13, 2nd edition, Carleton Univ., Ottawa, Canada. 2.

, (1980), Brownian motion, (in Japanese)

Iwanami Pub. Co. 1975.

English t r a n s l a t i o n , Springer-Verlag. 3.

, (1980), Causal calculus of Brownian functionals, and i t s applications.

4.

Proceedings of Ottawa Conference on S t a t i s t i c s & Related Topic

, Causal calculus in terms of white noise, Quantum Fields-Algebras, Processes, ed. L. Streit, 1-19.

5.

I. Kubo and S. Takenaka, (1980), Calculus on Gaussian white noise I, II. Proc. Japan Acad., 56 A, 376-380, 411-416, III (to appear).

6.

H.-H. Kuo, Fourier-Wiener transform on Brownian functionals.

7.

P. Levy, (1951), Probleme concrets d'analyse fonctionnelle.

(to appear). Gauthier-

Villars, Paris. 8.

P. Masani and N. Wiener, Non-linear prediction.

Probability and statistics:

The Harald Cramer volume (ed. U. Grenander), 190-212. John Wiley & Sons.

Almqvist & Wiksell,

74 T. Hida

130 9.

L. Streit and T. Hida, Generalized Brownian functionals and Feynman integrals,

(to appear in Stochastic Processes and their Applications).

75

GENERALIZED GAUSSIAN MEASURES Takeyuki Hida

§0.

Introduction When we discussed an application of white noise analysis to the "Feynman path

integral (see [8],[9]), we were naturally led to discuss generalized Gaussian measures on the space of paths and delta-functions of Gaussian random variables. The purpose of this short paper is to give a further interpretation to those generalized Gaussian measures in connection with other notations In white noise analysis that have been established recently. Before we come to rigorous description of the problem, it would be helpful to illustrate a naive idea behind our approach. tion, then

{B(t)}

with

B(t) = -r- B(t)

analysis deals with functionals of a member of variables. bles

R1'.

f f (t)B(t) dt

{B(t)}

be a Brpwnian mo-

B(t)'s, where each

The white noise

B(t) is thought of as

As a continuous analogue of functions on

x., l
[0,1] or

Let

is the white noise.

f(B(t), teT) , T

R

with varia-

being a parameter set, say

We can therefore introduce quadratic functional such as although a suitable renormalization is necessary so that it can

be viewed as a (generalized) functional. could be introduced.

Even Gaussian kernel

exp[-i.f B(t)2dt]

In fact, applying renormalization, Gaussian kernels as

well as Gaussian measures can be defined in the generalized sense. Section 1 is devoted to a quick review of the white noise analysis with much emphasis on generalized Brownian functionals.

Special properties of exponentials

of quadratic (generalized) functionals are summarized in Section 2.

In the last

section we shall introduce a class of generalized Gaussian measures. Part of the results in the unpublished paper [4] is reproduced in the present paper. As the author proposed there, he has hopes that the present discussion would be a contribution to the application of the white noise analysis to the theory of path integrals.

§!. GtYioAaLizzd Btwwntan

junctionati

Start with a Gel'fand triple E c L2(T) c E*, Suppl. Ren. Circolo Mat. Palermo, Serie II, No. 17 (1987)

76 230

TAKEYUKI HIDA

where of

T

R1*

i s a time p a r a m e t e r space t a k e n t e be a f i n i t e or i n f i n i t e The p r o b a b i l i t y d i s t r i b u t i o n of w h i t e n o i s e

measure

~\i on

interval

is a probability

E*, t h e c h a r a c t e r i s t i c f u n c t i o n a l of which i s given by

C(5) = e x p [ - J | l ^ l l 2 ] , ? e E.

(1)

{B(t)l

A member.of t h e complex H l l b e r t s p a c e

||

|\ : the

(L 2 ) = L 2 (E*, y)

L 2 (T)-norm. i s c a l l e d a Brownlan

functional. (L 2 ) (or a Fock space) i s t h e d i r e c t sum

The W i e n e r - I t o decomposition of

d e c o m p o s i t i o n . i n t e r m s of t h e space of t h e m u l t i p l e Wiener i n t e g r a l s

(L2) = I

(2)

H , n£0,

<&H..

n=0

The

T-transform,

d e f i n e d by

(3)

e1<x,C>

(T*) (5) = j

•(x)du(x),

E* ( L 2 ) - f u n c t i o n a l s t o t h e r e p r o d u c i n g k e r n e l H i l b e r t space fl? w i t h

carries kernel

C(E.-n), ( 5 , ri) e ExE.

The i n t e g r a l r e p r e s e n t a t i o n i s given by t h i s

T - t r a n s f o r m in t h e f o l l o w i n g manner.

Thzoiem Let

*

be in

H .

Then

(4) where

(T<(,) (?) = i C U ) U ( S ) , U(E)

i s e x p r e s s e d i n t h e form

(5)

with. * .symmetric

U(5) = | . . .

un)?(Ul)...5(un)dun

| F(m

L2(Tn)-function

F

(notation: F e L2(Tn) = L

satisfying (6) Such a f u n c t i o n

| | * | | { L 2) = • £ ! F

i s unique.

||FllL2(Tnv

(lfn)

77 GENERALIZED GAUSSIAN MEASURES

The f u n c t i o n

F

231

in t h e above theorem i s c a l l e d t h e k e r n e l , and

(4) i s t h e U - f u n c t i o n a l a s s o c i a t e d w i t h

4 £ H .

For a g e n e r a l

U(£)

in

4 e (L 2 ) we

f i r s t have t h e decomposition

4 = 7 4 , 4 e H , then t a k e U - f u n c t i o n a l s U (£) TI n n n n a s s o c i a t e d w i t h 4 . The sum U(£) = 7 U (£) i s a s s o c i a t e d w i t h 4 . n '•n n There i s a n o t h e r way t o d e f i n e t h e U - f u n c t i o n a l f o r a g e n e r a l 4 e (L 2 ) (Kubo and Takenaka) by u s i n g t h e S - t r a n s f o r m

(S*)(E) = C(5) | e < x , ? > < t > ( x ) d y ( x ) .

(7)

What are stated in the above Theorem may be expressed shortly

A H n = L 2 (T n )

(8)

(up to ȣT).

Now we have the idea to introduce a c l a s s of generalized Brownian functlonals igree of degree

n, which iiss an extension of

H .

The technique can be i l l u s t r a t e d by

the following diagram. H (-n)

-

X

&

= LJ(Tn)

Hn

where m

^(n+D/2^

H (T ) consists of symmetric functions in the Sobolev space of order

over

T n . The space

h^

involves test functlonals, while

H

is

the space of generalized Brownian functlonals of degree n. To introduce the entire space of generalized Brownian functlonals, we first choose positive increasing sequence

CL*>+ = {* = I

where

|| ||

is the

H

{c } and

v •«.«"» M ^ J I * -norm.

The space

test functlonals, and its dual space Brownian functlonals.

2

-

(L2)

is called the space of

( L ) is called the. space of generalized

78 232

TAKEYUKI HIDA

Remark

With- the measure u almost a l l x in E* is viewed as a sample function of B. Therefore, we often use the symbol B in place of x to have i n t u i t i v e expressions. For example <x,E> is often denoted by or by B(E;)- Note that B(t) is not a generalized functional but i t s renormalization:B(t) n : i s a member in H n §2. Expomntial

junctions

We are interested in exponential functions of quadratic, generalized Brownian functionals.

More precisely, given a formal expression such that

exp[c I B(t)2dt],

(9)

c < i.

T

I t can be made to be a generalized Brownian functionals by applying the multip l i c a t i v e renormalization. In the notation established in the previous section, we are given a generalized Brownian functionals

<>j (x) = :exp

(10)

x ( t ) 2z d t ]

[c I

C

W

The associated U-functional i s expressed in the form

(11)

U (E) = exp[c'

C(u) z du],

To fix the idea, the time parameter set T T = [0,1]. Let

l-2c

is taken to be the unit interval:

Several interesting properties are now in order. \J)(x) be a test functional of the form

f 1 f1 (12)

c'

*(x) =

"f/2

F(u,v):x(u)x(v):dudv, 0

F e H

0

and let

F(u> v) = I — n n ^ n n ( v ^

([0,1]2)

79 GENERALIZED GAUSSIAN MEASURES

be t h e H i l b e r t - S c h m i d t expansion of t h e k e r n e l f u n c t i o n

233

F.

I t i s c e r t a i n l y of

trace class. Piopoi-Ltion Let i)

7 <> (

and

ij1 be given by (10) and ( 1 2 ) , r e s p e c t i v e l y .

t h e p r o d u c t g i v e s t h e t r a c e of


where

< , >

F

up t o t h e c o n s t a n t

Then

2c':

c >y = 2 c ' Ly \~>' n n

T

i s t h e c a n o n i c a l b i l i n e a r form c o n n e c t i n g

(L 2 )

and

(L 2 )

,

(-2) be t h e o r t h o g o n a l p r o j e c t i o n of <> ( down t o H . Then, t h e c (-2) sum 4>I+I|J i s a g a i n an H - f u n c t i o n a l whose k e r n e l f u n c t i o n has e i g e n v a l u e s c ' + l / A , n = 1, 2, . . . ii)

let

$.

PfLOOJ

These two associated with

a s s e r t i o n s a r e almost o b v i o u s , if we see t h a t <(>i

U-functional

i s of t h e form c'

I C(u)2du, T

which a r i s e s in t h e Taylor s e r i e s expansion of We remind t h a t if

3

is the d i f f e r e n t i a l

U

in ( 1 1 ) .

operator (see, e.g.

[5]),

a c t i o n i s r e p h r a s e d a s t h e F r e c h e t d e r i v a t i v e a c t i n g on t h e a s s o c i a t e d tional.

In s h o r t , 3

t**-*6??tT

provided

$4—)VS.

The Levy Laplacian

(13)

u

A

can by given by

AL =

Simple computation tells us that for

3 2 (dt) 2

U(0

(Kuo's notation).

given by (11) we have

its

U-func-

80 234

TAKEYUKI HIDA

6

«5(t)

U (?) = 2 c ' 5 ( t ) U (?) = 2 c ' w

c

" "" < ^ ' " c ^ '

^

[ C(u)S (u)du-U (?)

J *^'"t

and

^ g ^ U ^ O Hence, we have t h e f o l l o w i n g

=2c«Uc(0^

(«t(t)=i).

proposition.

?K0p0i>MMm 2 (P. Levy [ 1 ] ) The e x p o n e n t i a l f u n c t i o n (14)

<>j

i s t h e e i g e n f u n c t i o n of t h e Levy L a p l a c i a n :

ALc = 2c'4> c . J.Potthoff

[ 7 ] has i n t r o d u c e d t h e n o t i o n of a p o s i t i v e g e n e r a l i z e d

in t h e f o l l o w i n g manner.

A member 2

(L )

iji

in

Let

(L2)~

2

CL ),

functional

be t h e c l a s s of p o s i t i v e ( t e s t ) f u n c t i o n a l s .

i s s a i d to be p o s i t i v e if and o n l y i f

$

maps

i n t o t h e p o s i t i v e r e a l numbers. We can o b v i o u s l y prove

Viopoi-Uxon 3 G e n e r a l i z e d Brownian f u n c t i o n a l s

<>j , c < i , a r e p o s i t i v e .

F i n a l l y , we have

Pfuopoiltlon 4 The e x p o n e n t i a l f u n c t i o n

fi>

following functional d i f f e r e n t i a l

3

given by (10) i s c h a r a c t e r i z e d by t h e equation:

t*c =

2 c

'3tV

(15) < l , * c > = 1, where §3.

3*

i s t h e a d j o i n t o p e r a t o r for

GunzKoLLzzd. Gauiila.n Let

T = [0,1]

d e f i n e d by ( 1 0 ) .

The

3 .

mea&uhZA

and l e t

<>j (x)

T - t r a n s f o r m of

be a g e n e r a l i z e d Brownian $ (x)

i s g i v e n by

functional

81 GENERALIZED GAUSSIAN MEASURES

1/2 (T
(16)

235

C(u)2du], c < i

This may be understood to be the characteristic functional of the Gaussian measure with variance case

YZ5—

» provided -°» < c < J.

Hence, we should avoid the

c > 1/2. Now set

(17)

dvc(x) = <)>c(x)dp(x).

Then, the T-transform of



may be thought of as the Fourier transform of

dv (x), since

(T* )(?) = [ e 1<X ' C> 4 (x)dy(x)

i<x,£>, , , 5 dv (x)

and since the expression (16) tells us that is known that

$ (x)

is positive.

dv

looks like Gaussian.

We therefore call

dv

with

Also it

c
a

generalized Gaussian measure on E*. It is interesting to note that the generalized Brownian functionals (which look like, as it were, the densities

<(> (x)

dv /dv) are eigenfunctions of the

Levy Laplacian with negative eigenvalues

2c'

2c

i

l-2c

REFERENCES [1] LEVY P.

Problemes concrets d'analyse fonctionnelle, Gauthier-Villars 1951.

[2] HIDA T.

Brownian motion, (in Japanese), Iwanami, 1975; English ed.

Applications, of Math. 11, Springer-Verlag 1980. [3]

Analysis of Brownian functionals, f.arleton Mathematical Lecture Notes, 13, 1975, 2nd ed.1978.

[4] [5]

Generalized Gaussian measures, preprint, 1983. Generalized Brownian functionals and stochastic integrals. Appl. Math. Optim. 12 (1984), 115-123.

82

236

TAKEYUKI HIDA

[6] KUO H.-H.

Brownian functionals and applications, Acta Applicandae Math. 1

(1983), 175-188. [7] POTTHOFF J. Positive generalized functionals, preprint (1986). [8] STREIT L. and HIDA T. integral.

White noise analysis and its application to Feynman

Lecture Notes in Math. no. 1033, Springer-Verlag, Measure

theory and its applications. [9]

,

(1982), 219-226.

Generalized Brownian functionals and the Feynman

integral. Stochastic Processes and their applications, 16 (1983), 55-69. [10] YOKOI Y.

Private communication

TAKEYUKI HIDA DEPARTMENT OF MATHEMATICS NAGOYA UNIVERSITY CHIKUSA-KU, NAGOYA 464, JAPAN

on positive generalized functionals.

83

The Impact of Classical Functional Analysis on White Noise Calculus* TAKEYUKI HIDA

Department of Maathematics, Meijo University, Tenpaku, Nagoya 1^68, Japan

§0. Introduction The remarkable results of functional analysis developed in Europe by V. Volterra, J. Hadamard, R.M. Frechet, P. Levy and others for the period from late 19th century to early this century have now received a new understanding and their valuable contribution is re-evaluated from many viewpoints. Our aim is to review the classical functional analysis from our standpoint of the white noise analysis with special emphasis on the part of infinite dimensional harmonic analysis. Such a consideration will discover a new bridge connecting the two kinds of analysis and will further give us problems to be investigated. To concretize the story the basic Hilbert space H, on which the classical analysis is discussed, is taken to be L2\0,1] or L 2 (5 1 ). We are interested in the analysis of (nonlinear) functionals on H; namely, we discuss differential and integral calculus for functionals of the form (p(x), x € H . Observe now the variable x £ H. It is viewed as (0.1)

x(t) = xt,

te[0,i].

So that x is, formally speaking, a vector living in continuously infinite dimensional vector space. In fact, such an understanding is fitting for many applications, where each xt is a variable and (0.2)


would be a favorable expression. We then come to a differential operator, which is considered in the usual manner. It is necessary to note the domain together with the topology equipped with it, when the Gateaux derivative, the Frechet differential and related operators are considered. As for the integrals of the functionals on function spaces, many attempts have been made so far in introducing suitable measures and in choosing a class of integrands. Even now questions arising from this topic are raised and we shall find good problems in this field. *Centro Vito Volterra, Univ. degli Studi di Roma II, March 1992, No. 90, pp. 1-20

84 §1. Motivation (I) First we consider integration over an infinite dimensional space H. Taking the topological Borel field B(H), the measurable space (H, B{H)) can easily be formed. Then, a difficulty arises, since it is impossible to introduce a Lebesgue-like measure on H. However, we can find ideas in the classical functional analysis; namely they have introduced the notion "la valeur moyenee" and "le passage du fini a l'infini" in order to carry out integration. A modernized version of their ideas may be seen in the white noise analysis. This is not an accident, because ideas behind the formulation should be similar at least in spirit. This is exactly what we aim to explain first. Actual story starts with measure. We are going to introduce an ideal measure on the unit ball S in the infinite dimensional vector space H: S = {z; \z\ < 1} ,

| •|

being the .fiT-norm.

Note that the unit sphere and the unit ball are not discriminated in an infinite dimensional case. We therefore approximate \x\ — 1 by

(1.1)

X>?- = 1

oi

Sn-l(V^)-

i=i

We are now ready to think of the probability measure on the sphere invariant under the rotation group SO(n). So long as n is finite, such a measure is nothing but the normalized measure proportional to the surface element. Then, we let n tend to oo, and we try to find finite dimensional analogue as much as possible. We obtain the so-called hyperfinite analysis over 5°°(-v/oo) with rotation invariant probability measure, i.e. the Gaussian measure fi (or white noise measure). Now, it should be emphasized that if we revisit the classical functional analysis and review it carefully, then we certainly find that they have discovered more profound results, although there are naive ideas and expressions are often intuitive. They have known really infinite dimensional objects which cannot be obtained by taking limits of finite dimensional stories. We are now ready to explain the diagram at the end of this paper. The symbol n describing the dimension is changed to d. The analysis over Sd using the rotation group SO(d) is well known, so it is put on the basement B. Let d tend to infinity to go up to the first floor I. Many of the results are obtained as the limit of the facts that can be seen in the basement, and they are now quite popular. For instance, the direct sum decomposition of (L 2 ) = L2(fi) into the space of multiple Wiener integrals (or homogeneous chaos), irreducible unitary representation of the infinite dimensional rotation group Ooo (defined later, in reality we use only a subgroup of O^ which is relatively small), infinite dimensional Laplace—Beltrami operator determined by the Ooo, and so forth. (II) We are now in a position to tell how and why we have climbed up to the second and the third floors in the diagram. We must quote Paul Levy's words from his autobiography (1970): "J'ai bati un etage de cet edifice; que d'autres continuent!". So, let us succeed! The guiding principle is based on the choice of a subgroup of Ooo to carry on the analysis.

85 The first floor has been constructed by using the subgroup G ^ : (1.2)

Goo = ind. limit of Gd,

Gd^

SO{d).

In the similar manner we can form floors II and III by taking subgroups as we shall show in what follows. §2. W h i t e Noise Integration with respect to the white noise measure realizes the trick to have la valeur moyenne on H. The measure does not exist on H, but does exist on a space of generalized functions, which is an extension of H. We now follow the steps to introduce the white noise measure. Take a nuclear space E such that a Gel'fand triple (2.1)

EcHcE*

is formed. Given a continuous, positive definite functional C(£) = exp [—11£|2]. Then, there exists a measure /J, on E* such that

(2.2)

G(0 = y"exp[i(z,0]<W-

The measure space (E*,/z) is the white noise, and fi is the white noise measure. Integration of functionals by \i can be compared to the la valeur moyenne as was mentioned before; this can be recognized by the Gateaux formula that gives an expression of the mean value of a functional

x(tn)),

then, the mean value is (2.4) (27r) _ n / 2 / • • •


,xn)exp[-(xl+xl+-

• -+xl)/2]dxidx2

• • -dxn .

§3. Rotation Group To claim that the white noise measure /z is rotation invariant, the rotation group has to be rigorously defined. Before its definition is given, we have to introduce the Levy group. Take a complete orthonormal system {£ n }- Assume that an automorphism •K of positive integers satisfies the condition (3.1)

lim — # { n < N; n(n) > N} = 0,

#{•} : number of sets {•} .

Further assume that the transformation g^: (3-2)

9* : £ = ^

a

n€n -> 9A

= ^2

"n^n)

is a continuous linear isomorphism of E. The collection of such g^'s forms a group called the Levy group and is denoted by Q.

86

Each gn exchanges the coordinates, possibly infinitely many £ n 's at the same time, but restricted so as to hold (3.1). REMARK 1. The left-hand side of (3.1) is often called the density of n. REMARK 2. The Levy group is a discrete group. It is covenient for us to take a group generated by Goo and Q. Here is an important remark. There are ways to show that there are many g^ (better to say general gn) which cannot be approximated by finite or hyperfinite dimensional rotations. Yoshizawa's definition (1969) of rotation group is as follows: DEFINITION. If a linear transformation g acting on H satisfies 1) it is an isomorphism of E and 2) it is an orthogonal transformation of H, then g is called a rotation of E. The collection of the rotations of E forms a group. We call it the rotation group of E and denote by O(E). If the basic nuclear space is not specified, we call the group the infinite dimensional rotation group and denote it by OooREMARK 3. The above definition is independent of the choice of a complete orthonormal system in H. Let {£ n } be a complete orthonormal system in H such that £„ G E for every n. Denote by Ed the subspace of E spanned by the first d £„'s. Ed is isomorphic to Rd. Now set Gd = {g e 0{E);g\Ed

G SO(d), g is the identity on Ej-} .

Obviously Gd is a subgroup of 0(E) and is isomorphic to SO(d). If we take the inductive limit of Gd, then we are given the hyperfinite subgroup GQO mentioned in §1. Again, observe the diagram. The basement B is the place where the finite dimensional game is played by using SO(d) and Sd. Floor I provides the analysis over S°°(y/oo) with Goo-invariant measure /u. The results such as the direct sum decomposition of (L2) in terms of Hn's involving homogeneous chaos of degree n, irreducible unitary representation of Ooo or Geo is given on each Hn, determination of the infinite dimensional Laplace—Beltrami operator in terms of GQO , and so forth, can be obtained mostly by taking a limit of those in the finite dimensional case. As a development of this story we can even discuss the analysis in the hyperfinite case. We wonder why the people of the classical functional analysis did not pay much attention to these directions. Returning to the rotation group Ooo, we consider the rotation invariant property of the measure fi. Associated with g £ 0^ is a transformation g* of the space E* which is uniquely determined by the relation

(3.3)

(g*x,0 = (x,gO,

xcE*,

ZeE.

The product g*n defines a new measure determined by the characteristic functional C(g£). It is in agreement with C(£), so that we have (3.4)

g*fi = /z.

_ _

87

This property enables us to have a unitary representation of the rotation group O^; we have already used this fact. We now understand that first floor in the diagram can be constructed in a natural manner and play the role of the foundation of white noise analysis. Although we can still see interesting questions and connections with applications, first floor may be called an established area. §4. Generalized Functionals We are now on the second floor. When a generalized white noise (or Brownian) functional was introduced, the idea came from very simple requirement. Namely, the white noise measure is nothing but the probability distribution of the time derivative {B(t)} of a Brownian motion {B(t)}. REMARK. In view of this fact, {B(t)} itself is often called a white noise. Therefore, /x-almost all x(t) in (0.1) are viewed as sample paths of B(t), and most typical example of a functional
(S)C(L2)C(S)*,

where the test functional space (5) is formed by the second quantization technique. The space (5)* is the best choice for the space of generalized functionals for our

calculus, in particular for the second floor in the diagram. Some examples are now in order. 1) Normal functional. It is defined by the S-transform: (4.2)

um)£(uiyt(u2)k

U(0 = J ••• J F(uuu2,...,

•••

i{um)ldu^k+-"+'.

Note that the Potthoff-Streit criterion guarantees that the above U is the Stransform of an (S) "-functional. 2) Gauss kernel.
(4.3)

c ,U/2.

3) J-function. (4.4)

ip(x) = S(B(t) — a),

B being a Brownian motion.

The Levy Laplacian A^ is characterized by the Levy group. It is given by the following formula: (4.5)

A t = lim — Aw ,

A^ the ./V-dimensional Laplacian.

N JS

The A i vanishes on (L2), but it acts effectively on the space (S)*. The classical analysis has recognized that A^ is a derivation. §5. Causal Calculus Another interesting encounter of the classical analysis and the white noise analysis is the differential operator, although expressions are somewhat different. Consider the operator (5.1)

ftS^

(or

=8/86(1)).

This is rigorously denned by using the S-transform and the Prechet derivative. More precisely, for the S-transform U(£) of the given
S-\U'(t,t))(x)=dt
where the Frechet derivative U'(£, t) is determined by the so-called Volterra formula: (5.3)

SU=

[u'(Z,t)6Z(t)dt.

The adjoint operator 9t* of dt is denned by the canonical bilinear form ( , )M that connects (S) and (S)*: (&^) =

foWMI

v& (S),

Ve(S)

89 Again, note that the suffix t of these operators stands for the time. They are fitting for the causal calculus. The infinite dimensional Laplace-Beltrami operator AQQ can be expressed in the form

(5.4)

Aoo = -

Jd&dt,

while the Levy Laplacian is expressed in a formal form as follows: AL = !{dtf{dt)2

(5.5)

(H.-H. Kuo).

Simple observation shows that the group G^ determines the rotations of the function space with the coordinate system {x(t), t e R} (see the Volterra formula). We can therefore introduce generator jt,s of the rotations on the two-dimensional (x(i),x(s))-plane: (5.6) There are Laplacian There where the

7 M

- d*ds -

a;dt.

many roles played by these generators, among others a determination of operators. are many applications where the time parameter t is emphasized and causal calculus is applied. Here are typical examples of application.

1) Feynman's path integral. 2) Dirichlet forms with continuously many degree of freedom. Constructive quantum field theory. 3) Mathematical description of the movement of primitive animals interfered with by some "fluctuation". So far we have had an overview of the contents in the second floor II. One can further discover interesting objects there often suggested by the classical analysis. However the third floor is now more fascinated to us. §6. Whiskers Let us go up to the third floor which is an unexplored field. One is directed to this place by a new subgroup of OQORemind that the subgroups G^ and the Levy group Q are both defined with the help of a complete orthonormal system {£ n }. One may ask if there is any subgroup given without use of any {£ n }- Also think of the particular role of the time shift in the casual calculus. Thus we come to a member of 0{E) defined by changing the time parameter. To be general, the time variable u is assumed to be a vector in Rd. Then, consider a g such that (6.1)

ff:««)->(sO(«)

= W(«))V^7Wl.

where tp is a diffeomorphism of Rd and ip' is the Jacobian. Such a transformation is too general to find analytic structure and hence, by the suggestions of the actual problems, we come to one-parameter subgroup of 0^ consisting of such g's as in

90

(6.1). One-parameter group property gtgs — gt+s or that of ipt specifies the structure of the group {gt} more than we expected. DEFINITION. A continuous one-parameter subgroup of 0(E) consisting of g's given as in (6.1) is called a whisker. Important examples of a whisker are listed below. 1) 2) 3) 4)

shifts S% : £(u) —> £(u — tej), ej j t h coordinate vector, dilation r t : £(u) -> i(etu)etdl2:, rotations on i? d : SO(cO, special conformal transformation: fcj = uSfuj,

where w is the reflection with respect to the unit sphere. These four subgroups generate a {d(rf + 3)/2 + l}-dimensional subgroup of 0(E) which is a Lie group locally isomorphic to SO(d + 1 , 1 ) . The subgroup is denoted by C(d) and is often called a conformal group. We are now interested in the action of the conformal group or in the field on the third floor where the group plays. Once again, the classical functional analysis gives us suggestions to discuss functional U(£) that depends on surface C with unit codimension, like in the cases of the Hadamard equation for Green's function, equations for charged surface moving in an electro-magnetic field, and others. Within the framework of the white noise analysis, we can think of a random field X(C) depending on a parameter C of (d — l)-dimensional manifold. Assume that it lives in (S)*, so that we may write X(C,x) and discuss its variation SX(C) when C deforms. If we use the S'-transform, there appears an equation for U(C,£), so that we can appeal to the classical theory. To fix the idea, we assume (6.2)

X(C,g*x)=X(g'C,x),

for

g e C(d),

where g' is a transformation of u determined by g and the mapping from g to g' is bijective. An obvious example is an X(C) obtained by white noise integral over the domain encircled by C or its function. Deformation may be assumed to come from the transformation on Rd that is in C(d). With these assumptions we are given a stochastic variational equation of the form (6.3)

SX(C) = (X, C, {ak},

{dtk}),

where a^s are infinitesimal generators determined by a basis of the Lie algebra of C(d). In terms of the S-transform U(t,), (6.3) implies (6.4)

SU = /(£/, C, 5n), 5n displacement along normal to C .

If, in particular, the right-hand side is expressed in the Volterra form, many results have been obtained. Most interesting example is Tomonaga's generalization of the Schrodinger equation which gives a solution of wave function depending on C, that is the super many time theory. "J'ai bati un etage de cet edifice; que d'autres continuent!" - from Paul Levy's autobiography, 1970.

91

[Diagram] Infinite dimensional Harmonic Analysis - arising from Ooo III. Ultra infinite dimensional case {X(C)} random field living in (S)* -> U(C) = U(C,Z) Variation: Electromagnetic fields, Hadamard equation Super many time theory {ip(C)};

diffeomorphism ip —»• g whisker {ipt} ->{&}, 9s9t = 9s+i, whiskers get innovation; conformal group C(d) deform C to have stochastic variation. II. Infinite dimensional case

(S) c (i2) c (sy

Potthoff-Streit characterization of (5)* H~ C (S)*, Hermite polynomials in x(t)'s Levy group Q -> Levy Laplacian A^ = f d2{dt)2 Laplacians AQ, A V : Obata's characterization Gauss kernel Nexp [— ^ JTx(t)2dtj: eigenfunctional of A/, Kuo's Fourier transform on (S)* Classical theory for (L2)~ I. Hyper finite dimensional case (L2) = ®Hn, extends to hyper functional supp (/i) ~ 5 ° ° ( v ^ ) Hyper finite group G^ ~ limG n , Gn ~ SO(n) unitary representation of G^ on Hn Laplace—Beltrami operator Aoo = J2n \ (jS~ ) — (x, £„) a

3£n

—AQO = N: number operator Hn: eigenspace of Aoo, eigenvalue: —n irreducible unitary representation of Goo on Hn Fourier-Wiener transform B. Finite dimensional case L2(Sd) = ®H„, Sd ~ SO(d + l)/SO(d) hyper functions on Sd Group SO(d + 1), Ad'- spherical Laplacian Unitary representation of rotation group Fourier series The analysis in each storey (except Basement B) above corresponds to a subgroup of Ooo.

92

References (for further reading) [Books] Vito Volterra (1860-1940) [VI] Leons sur les fonctions de linges. Gauthier-Villars, 1913. [V2] (with J. Peres) Theorie generale des fonctionneles. Gauthier-Villars, 1936. Livre I, Generalites sur les fonctionelles. [V3] (with B. Hostinky) Operations infinitesimales lineaires. Gauthier-Villars, 1938. [V4] Opere matematiche di Vito Volterra, volume quinto. 1926-1940, Accademia Nazionale dei Lincei, 1962. [V5] Theory of functionals and of integral and integro-differential equations. Dover Edition, 1959. Jacques Hadamard (1865-1963) [Hi] Leons sur le calculus des variations. Herman et Fils, 1910. [H2] The psychology of invention in the mathematical field. Princeton Univ. Press, 1945. [H3] (Euvres de Jacques Hadamard, Tome II, CNRS, 1968. Rene Maurice Frechet (1878-1973) [Fl] Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo, t. 22 (1906), 1-74; Les ensembles abstracts et le calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo, t. 30 (1910), 1-26, Japanese transl. by Saito et al., 1987. Paul Levy (1886-1971) [Ll] Les equations integro-differentielles definissant des fonctions de lignes, 1-120. Theses. Docteur Sci. Math. 1911. [L2] Leon d'analyse fonctionnelle. Gauthier-Villars, 1922. [L3] Analyse fonctionnelle. Memorial des Sciences Mathematiques, # 5, GauthierVillars, 1925. [L4] Cours de mechanique. Gauthier-Villars, 1928. [L5] Problemes concrets d'analyse fonctionnelle. Seconde ed. des [L2], GauthierVillars, 1951. [L6] Analyse fonctionnelle. Fonctions de lignes et equations aux derivees fonctionnelles. Gauthier-Villars, 1951. [L7] Random functions: General theory with special reference to Laplacian random functions. Univ. of California Pub. in Statistics vol. 1, no. 12. (1953), 331-390. [L8] Quelques aspects de la pensee d'un mathematicien. Albert Blanchard, 1970. Other references 1. Selected papers on quantum electrodynamics. Ed. J. Schwinger, 1958, Dover Publ. 2. R. Courant, Dirichlet's principle, conformal mapping, and minimal surfaces. Interscience pub., 1950; Repr. Springer, 1977. 3. S. Tomonaga et al, Quantum field theory (in Japanese) Iwanami.

93 [Papers] V. Volterra [v.l] Sopra le funzioni che dipendono da altre funzioni. Rend. R. Accad. dei Lince. (I) vol. Ill (1887), 97-105; (II) 141-146; (III) 153-158. See "Opere Matematiche", vol. 1, 294-314. [v.2] Drei Vorlesungen ueber neuere Fortschritte der Mathematischen Physik. Archiv der Mathematik und Physik, III Reihe, Band XXII, Heft 2/3, (1914), 95-181. J. Hadamard [h.l] Sur quelques questions de calcul des variations. Ann. Ecole Norm. Sup. 24 (1907), 203-231. [h.2] Le developpement et le role scientifique du calcul fonctionnel. International Congress, Bologna, 1928. P. Levy [1.1] Sur la fonction de Green ordinaire et la fonction de Green d'ordre deux relatives au cylindre de revolution. Rend. Circolo Mat. Palermo, 34 (1912), 187-219. [1.2] Sur la variation de la distribution de I'electricite sur un conducteur dont la surface se deforme. Bull. Soc. Math. France, 46 (1918), 35-68. [1.3] La mesure de Hausdorff de la courbe du mouvement brownien. Giornale dell'Istituto Italiano degli Attuari, 16 (1953), 1-37. [1.4] Jacques Hadamard, sa vie et son (Euvre calcul fonctionnel et questions diverses. La vie et l'Oeuvre de Jacques Hadamard, 1967, 1-34. [1.5] Fonctions de lignes et equations aux derivees fonctionnelles. 18e Congres international d'histoire des Sciences, 1971, Moscow. S. Tomonaga [t.l] On a relativistically invariant formulation of the quantum theory of wave fields. Prog. Theor. Phys. vol. 1, no. 2 (1946), 27-42. P.A.M. Dirac [d.l] Relativistic quantum mechanics. Proc. R. Soc. London, A. 136 (1932), 453-464. Cf. many time formalism. F. Dyson [d.l] The radiation theories of Tomonaga, Schwinger, and Feynman. Phys. Rev. 75 (1949), 486-502. [d.2] The S matrix in quantum electrodynamics. Phys. Rev. 75 (1949), 1736-1755. M.D. Donsker and J.L. Lions [d-z.l] Frechet-Volterra variational equations, boundary value problems, and function space integrals. Acta Math. 108 (1962), 361-375.

94

T. Hida [h.l] White noise analysis and Gaussian random fields. Proc. 24th. Karpacz Winter School on Theoretical Physics, Jan. 1988. World Scientific, 277-289. [h.2] Brownian motion and Levys' functional analysis. Sugaku Seminar in Japanese, 1988. [h.3] Stochastic variational calculus. Proc. IFIP Conf. Charlotte, 1991, to appear. Si Si [s.l] Gaussian processes and conditional expectations. BiBoS Notes Nr. 292/87, Universitat Bielefeld, 1987. [s.2] Variational calculus for Levy's Brownian motion. Proc. Nagoya Conf. on Gaussian Random Fields, August 1990, World Scientific. [s.h.l] (with T. Hida), Stochastic variational calculus in white noise analysis, preprint. A. Noda [n.l] Levy's Brownian motion and stochastic variational equation. Proc. of the Nagoya Conf. on Gaussian Random Fields, August 1990. World Scientific. H. Yoshizawa [y.l] Rotation group of Hilbert space and its application to Brownian motion. Proc. Internat. Conf. on Functional Analysis and Related Topics (1970), 414-423.

MEMOIRS OF T H E COLLEGE OF SCIENCE, UNIVERSITY OF KYOTO, SERIES A

Vol. XXXIII, Mathematics No. 1, 1960.

Canonical representations of Gaussian processes and their applications By Takeyuki

HIDA

(Received April 2, 1960)

Introduction Let B(t), O^if-c^oo, be an additive real Gaussian process with E(B(t)) = 0 and F(t, u) be a real-valued function of (t, u). The process X(t) defined as (0.1)

X(t) = ['Fit, u)dB(u) Jo

is a real Gaussian process with mean 0, and enjoys the property (0.2)

Ttt(X) C 9K,(fi),

0 ^ t < oo ,

where Tlt(X) and W^B) denote the closed linear manifolds generated by {X(T) ; T <; t} and {B(r); r <; t) respectively. Given a Gaussian process X(t), P. Levy called the expression (0.1) a representation of X{t), if X(t) is version of X(t) and he introduced the concept of canonical representation. Roughly speaking, a canonical representation is one for which the equality holds instead of the inclusion relation in (0. 2) (cf. Definition I. 2 and Theorem I. 2). In this case, F{t, u) is called a canonical kernel. P. Levy has recently published several important papers concerning the canonical representation of Gaussian processes. However his pioneering works contain some points difficult for us to follow. The main aim of this paper is to establish his theory systematically and to prove some new facts. We shall here give a brief account of the contents of this paper. In this paper we shall treat only real Gaussian processes and often omit the adjective "real Gaussian",

110

Takeyuki Hida Section I. General theory.

As P. Levy proved in [4], 13 a canonical representation of any process is uniquely determined if it exists. We shall prove this fact in detail in §1.2. Further we shall give a necessary and sufficient condition for the existence of the canonical representation, using Hellinger-Hahn's Theorem in the theory of Hilbert Space. This fact is not found in Levy's paper. As to whether a given representation is canonical or not, Levy gave a criterion using Hellinger integral (P. Levy [6]). But we shall give another criterion which proves to be a generalization of Karhunen's kernel criterion for the moving average representation of stationary processes. Section II. Multiple Markov process. J. L. Doob (for stationary processes) and P. Levy defined Nple Markov processes2' using the derivatives up to the (N— l)-th order of the processes. We generalize this notion to treat the processes which are not always differentiable. The main results obtained here are as follows. The canonical kernel of the A^-ple Markov process is a Goursat kernel of order N. This generalizes the fact obtained by Levy. In § II. 3, we shall prove that the stationary A^-ple Markov process is the sum of special simple Markov processes which are to be called general Ornstein-Uhlenbeck's Brownian motions. This also generalizes Doob's Theorem [1]. Section III. Levy's M(t) process. Let X(A), A£EN (EN is iV-dimensional Euclidean space), be an ordinary Brownian motion with A^-dimensional parameter (cf. P. Levy [4]) and MJf) be the average of X(A) over the sphere with center O (origin of EN) and radius £(2^0) in the parameter space EN. MN(t) is clearly a Gaussian process with time parameter t. Levy discussed the canonical representation and the multiple Markov property for this process MN(t) only in the case N is odd. We shall simplify his proof by transforming MN(t) into a stationary 1) paper. 2)

Numbers in square brackets refer to the list of references at the end of the In Levy's terminology "Markov process of order iV in the restricted sense".

Canonical Representations of Gaussian Processes

111

process. Our present method is applicable to the case N is even. We shall prove that M2p(t) is not a multiple Markov process but the limiting process of multiple Markov processes. Furthermore we shall prove that there are 2P~1 different representation of Af2£-i(*), which provides an affirmative answer to Levy's problem (Levy [4, p. 146]). I would like to express my hearty thanks to Professor K. Ito for his encouragement and valuable suggestions and to Mr. N. Ikeda who helped me with valuable discussions in overcoming the difficulties in the course of this paper; in particular, the idea of using the reproducing kernel in the proof of Theorem I. 4 is due to Mr. Ikeda. Section I. General theory of representation. § 1.1.

Definitions.

In order to define a representation of the given Gaussian process precisely, it is necessary for us to consider integrals with respect to certain random measures. The parameter space T of a process that will be treated here may be a closed interval, (—°°, oo) or [0, oo). The symbol BT denotes the Borel field of subsets of T. Let B(M), Me BT, be a real Gaussian random measure such that (1.1)

E{B{M)) = 0 and E{B{Mf) = v(M)

for every

M&BT,

where v is a (non-negative) measure defined on BT. Then, B(M) can be decomposed into two parts in the following way B(M) = B1(M)+ S I < ; , where B-X-) is a random measure associated with the continuous measure v1(-) = E(B1(-)2) and Xt/s are mutually independent Gaussian random variables with mean 0, each one of which corresponds to the jump point ts of v(u) = v{{—oo, u~\r\T). .£?,(•) and {Xt3) will be called the continuous part and discontinuous part of B(-) respectively. Let f(u) be a Borel measurable function. Then, if feL2(v), that is, / 6 L ! ( » J and E/(fy) 2 £tXj .)<«>, then

112

Takeyuki Hida f{u)dBx(u)

and

E/tfAXiy

are well defined for any Borel set M. The integral of / over M may be written in the form (1.2)

( f{u)dB{u) = \ f{u)dBl(u)+ JM

'Ef(tJ)XtJ, tj^M.

JM

where the integrals which appear are interpreted in the usual way with respect to random measures dB, dBt, Doob [2; pp. 426-433.] Now we can give Definition 1.1. Let Y(t), t £ T, be a real Gaussian process with E(Y(t)) = 0 for every teT. Then the triple (dB(t), mt, F{t,u)), or simply the pair (dB{t), F(t, u)), is called a representation of Y(t), if i) 5(«) is a random measure satisfying (1.1) ; ii) F(t, u) is a real Borel measurable function of u vanishing for u^>t and belonging to L2(v) for every t; iii)

X(t) = (V(f, u)dB(u)»

is a version of Y(t); iv) Wit is the closed linear manifold generated by {X(r) ; r^,t}. The function F(/, M) is called a kernel of the representation. There are many examples of processes which have no representation. Furthermore, even if a process has a representation, it may not be uniquely determined. The following examples will serve to illustrate such circumstances. Example 1.1. Let Y^t) be a Gaussian process with covariance function T(s, t) = l for s=t, and = 0 for s=M, and with E(Y1(t)) = 0. Then Y^t) has no representation. Example 1.2. Let Btf) and £„(*), 0 < / < < ^ , be standard Brownian motions which are independent of each other. Define z( )

f Bt{t) ~ I 52(f)

Then Y2(t) has no representation. this will be given later.

if t is rational, if / is irrational. Detailed discussions concerning

3") The notation \ -"means the integral I

•••in the sense of (1.2).

Canonical Representations of Gaussian Processes

113

Example 1.3. Let B(t), §
m=l(c0+c^+c(fj+

-

n

+cn(*L)

)dBiu)

is again a standard Brownian motion (P. Levy [_&]). This proves that B(t) has infinitely many representations. We have now to determine the best class of representations for our purpose among all posible representations of a given process. Definition. 1.2. The representation (dB(t), SOt,, F(t,u)) is called canonical, if E(X(t)/Bs) = [SF(t, u)dB{u) holds for every s
= ( Fm(t,u)2dvm(u)

for every

M&BT,

JM

considering them as measures, where dv^{u) = E{dB«\u)*),

i = 1, 2 .

This relation obviously satisfies the equivalence relations and therefore we can get the classes of representations. Theorem 1.1. (P. Levy) For every Y(t), there exists at most one class of canonical representations. Proof. Let (dBCD(t), 5KJ", FCi\t,u)) i=l, 2, be canonical representations of Y(t). Writing Xc"(t) = (V co (f, u)dBCi\u), we have, for every t and every

i = 1,2,

s{^,t),

E((X«>(t)/B«>) = jV«>(*, u)dB«\u), where B\" denotes the Borel field corresponding to XCD(t). The equality

114

Takeyuki Hida E(E(Xm(t)/B™)2)

=

E(E(Xm(t)/B?>)2)

holds, since both sides are determined only by the probability law of Y(t). Hence we have \SFm(t, u)2dvm(u) = \SFm(t, ufdvm{u),

for every

s(
which proves the theorem. § I. 2. Canonical representations. In this article we shall study important properties of a canonical representation. Definition 1.4. A canonical representation (dB(t), SSlt, F(t, u)) is called proper if (1.3)

mt = W:t(B)

for every

t6

T,

where Tlt(B) is the closed linear manifold generated by

{

f

dB(u);

MeBT\.

(—oo,/]nM

Theorem 1.2. For any given canonical representation, we can construct an equivalent proper canonical representation. Proof. It is sufficient to consider the case in which v is a continuous measure. Let (dB(t), 501,, F(t, u)) be a given canonical representation of Y(t). We shall show that we can construct a proper canonical representation of Y(t) by deforming the given one. 1°) Deformation.

First define F(t, u) = F(t, u). Put

fj,{M) = V (\MFV> u)2dv(u)) ,<> v(M) = [ dv(u) = E([

dB(u))2,

MeBT.

Here we may suppose that v is a continuous measure.

Then

hence, by Radon-Nykodym's Theorem, there exists a Borel measurable function f(u) ~2z 0 such that 4) V means the lattice sum. 5) fi
Canonical Representations of Gaussian Processes

115

fi(M) = ( f(u)dv(u) JM

Since N=N(f)= {u ; /(w)>0} is Borel measurable, we can construct a random measure (1.4)

B{M) = ( dB(u) = \ %N{u)dB{u), JM

MeBT,

JM

where X^u) is the indicator function of N. 2°) The triple (dB(t), Wlt, F(t, u)) is a representation of Y(t). To prove this, it is sufficient to show that X{t) = \ F(t, u)dB(u) is the same process as Y(t), in the sense of equivalence in law. can be proved as follows. E(['F(t,

u)dB(u)-\'F{t,

E([F(t,

w)(l-%^(M))rfB(M)Y

(1.5)

This

u)dB(u)Y

u)2(l-XN(u)fdv(u)

\*F(t, [

(l~XN(u)YF(t,

ufdv{u) = 0 ,

(-co, f] n N

which shows that X(t) = X(t) (= ['F(t, u)dB(uyj with probability one. Since X(t) is the same process as Y(t) by assumption, the relation above proves our assertion. 3°) Finally we shall prove that the representation (dB(t), SSlt, F(t, u)) is proper canonical. Now we prove (I. 6)

mt ^

Tit(B).

s

Suppose an element Z of iJi,(S) is orthogonal to (hence independent of) Wlt, that is, E{Z-X{s)) = 0 for every s^t. Then Z is orthogonal to E(X(s")/Bs/) for every s'-gLt and every s", since it is an element of mt. On the other hand Z can be written as Z = Vh{u)dB{u), with a Borel measurable function h. Noting that

116

Takeyuki Hida

E(X(s")IBs>) = Projection of X(s") on 2tt,/ = Projection of X{s") on s$ls>

(in the U sense)

(since X(t) = X(t) with probability 1) = E(X{s")jBs,)

= T F(s", u)dB{u) (from canonical property)

= f F(s", u)dB{u), we have = \S h(u)F(s", u)dv(u) = 0 .

E(Z-E(X(s"))/B,,)) Therefore, for every s, s'(<^t)

I h(u)F{s", u)dV{u) = 0 . Since s" is arbitrary, we can prove MN(h)) = 0, where N(h)={u;

/z(w)#=0}. Hence we have

E(Z-W) = 0

for every

WeWlt(B),

Thus we have proved (I. 6). Consequently we have (I. 3). Now from (I. 3), E(X(t)/Bs) = projection of X{t) on 3H, = Projection of X{t) on SSlt(B) = (Vtf, «)rffi(«), which proves that the representation is canonical and (I. 6) implies that it is proper. Thus we have proved the theorem. By the argument used in the proof of the theorem, we hav Corollary. If a representation (dB(t), 2ft,, F(t, u)) (not necessarily canonical) satisfies the condition mt(B) = 3JI,

for every t,

then it is (proper) canonical. As is well known, a stationary process X(t) which is purely non-deterministic and M2-continuous can be expressed as

X(t) = T F(t-u)dB(u)

Canonical Representations of Gaussian Processes

117

and there exists one and only one representation having the property (1.3). (Karhunen [1]). In our case, in which X(t) is Gaussian, this means that it has a proper canonical representation. § I. 3. Existence of representation. In order to study the existence of the canonical representation we shall summarize some known theorems of the theory of Hilbert space. Let T(s, t), s, t £ T, be a real non-negative definite function. Then there exists a Hilbert space § satisfying i) r(s, t) belongs to § as a function of s, ii) = f(t) for every / e & iii) £> is the closed (in the topology || ||) linear manifold generated by {r(-,f); t&T}. r(s, t) is the reproducing kernel of that Hilbert space. The construction and the important properties of £> may be seen in Aronszajn £1]. We can construct sub-spaces £>, and £>? of £>: £>, = sub-space of £> generated by {P( •, T) ; T <^ t) ; £>? =

A£,+A.

Now let us assume that (H. 1) €> is separable, (H. 2)

A $, = {0}

(hence A $? = {0}).

Noting that

V7 £>? = £> and $ * C £ ? ,

*<',

we can see that there exists a resolution of the identity [E{t) ; t e T} such that (1.7)

$? = £(*)©,

by assumption. Then, by Hellinger-Hahn's Theorem,^ there exist two denumerable sets {fw}, i = l,2, ••• and ga)l, j , 1=1,2,— in § satisfying the following conditions (I. 8) to (1.10). 6) The symbol < , > denotes the inner product. norm, i.e. \\f\\=\/<J,7>. 7) For proof see M. H. Stone [ 1 ] or S. It6 [ 1 ] .

We shall use || || to denote the

118

Takeyuki Hida i) For any intervals Alf A2 < AJS/10, \Ef» > = 0, i=t=;, with AE=E(b)-E(a) for A = («, ft] ; ii) if A t AA 2 =0

(L8)|

= 0 ; iii)

for any i, />,(*)= IIEtf)/"3!!*

is continuous, non-decreasing and P.-+X/0,- (considered as measures); iv) g° 3 ' is the eigenvector of the self-adjoint operator H= \tdE(t) corresponding to the eigenvalue /_,(/= 1, 2, •••)• (1.9)

£> = m © 9?

(direct sum),

where 2Jt and Sft are denned by

2K(/"5) = { / ; / = \
5 /

,)};

),

yi(g°y!) = one dimensional sub-space generated by

a

(i. io)

co

gau;

E(tm(f >) c m/ ),

which is equivalent to (1.10')

E(t)Pi = P,E(t),

Pt = Projection on 2K(/">).

Furthermore, though there may be many ways of choosing such {fCD} and {g°:l}, their numbers are always the same. By virtue of this theorem, we can define the multiplicity of E(t). Definition 1.5. The supremum of the number of / c ' 5 's and the numbers of linearly independent eigenvectors corresponding to each tj is called the multiplicity of E(t). Theorem I. 3. T( •, t) is expressible as

(1.11)

r(., /) = 2 \'FM, u)dE(u)r°+ 2 2 b](t)g^<.

Proof. From (1.9), r(-, t) is written in the form

r(-, t) = 2 \Fi(t,u)dE{u)f^+ 2 2 b)(t)ga" •

Canonical Representations of Gaussian Processes

119

Applying (1.10), we have r(-, t) = E{t)T{., *) = 2 E(t) [F,(jt, u)dE(u)f°+ = 2 ^F,(t, u)dE(u)f°+

2 2

2 2 b){t)E{t)g^<

b){t)E{t)g^',

where the summation in the second term in the equation above extends over those g°'3/'s the eigenvalue of which are not larger than t. The case in which the function Ft(t, u) in (1.11) is degenerate, for example, d-12)

2/*(*)&<«),

is of special interest, as we shall see in the next section. After the preparation above, we can now discuss the existence of the canonical representation. Given a process Y(t), let r(s, t) be its covariance function. Let 2ft, be the closed linear manifold generated by {Y{r); r
We shall assume that (2ft. 1)

2ft is separable (as a sub-space of L2(Cl)),

(2ft. 2)

A 2 » # = {0}.

We shall prove the following preliminary theorem leading to the fundamental Theorems I. 5 and I. 6. Theorem 1.4. onto 9ft defined by (1.13)

There exists an isometric transformation from £> &3T{-,t)

«-»

Y{f)em.

This isometry induces the following correspondence: i) £>,~2ft„ £>?^2ft*, ii) E(t)f<^E(X/Bf) provided that f~X, with Bf =

BWt). 8)

Proof. Define a mapping S from L = {r(-, f); if e T} into 9ft by 8) { ••• } denotes the linear space generated by the elements that are written in the bracket.

120

Takeyuki Hida S:

r ( . , t) — Y(t) 2 « 2 « , m )

(«,-: real)

Then S is a linear transformation from a linear space L into the linear space 2= {Y(t), / e T } C ^ Suppose I>,TO-) = 0. Then 2«
/(*) = = o, if /(•) = 2 ««r(., o. This shows that S is a one-to-one mapping from L onto 8. further = V(s, t) =

And

E(Y(t)-Y(s)),

which proves that S is isometric. Since L and 8 are dense in §> and TO respectively, S can be extended to an isometric one-to-one linear transformation S from £> onto TO. Hence we have proved the existence of the isometry. Next, S£>, = TO, is obvious. Therefore if f*-*X, E(t)f *-» Projection of I on 1 * =

E(X/Bf).

Thus we have proved ii). Theorem I. 4 along with the assumptions (TO. 1), (TO. 2) implies that (H. 1) and (H. 2) hold. We can therefore appeal to the Hellinger-Hahn's Theorem. Let f(D be as defined in the statement of that theorem. Then there exists a continuous additive process Bco(t) such that dE(t)fw «-» dB«\t) under the correspondence mentioned in Theorem I. 4. Theorem 1.5. If Y(t) satisfies (TO. 1) and (TO. 2), there exist Gaussian random measures {Bcn(-)} and random variables Y\. such that i) Ba)(-), Y\., i, j , 1=1, 2, ••• are all independent, ii) E(5«+1>(.)')<£(fl">(-)2), iii)

* = 1,2,-,

t

Y(t) = Tl[ Fi(t, u)dB™(u)+ S S W i ' ! ,

iv) E(Y(t)/Bs)

= ^[SF(t,u)dB^(u)+^*^b)(t)Ylt.;^

9) 2 * 2 = 2 2 +

2

s
Canonical Representations of Gaussian Processes

121

Proof. If follows from (1.8), (I. 9) and Theorem I. 4 that i), ii) and iii) are satisfied. Considering the isometry S in the proof of Theorem I. 4, we have E(Y(t)/Bf)

= S-\E(s)T(., t)) = S-\E(s) 2 [FAt, u)dE(u)f™ + E(s) 2 2

By (1.10), E(Y(t)IBs)

b)(t)g^1).

is equal to

S_1(2 (V,-(f, u)dE(u)f™+ 2 * 2 &5W5') = Zi['Fi(t, u)dB™(u) +

-£*^b){t)Ylt. (E(X(t)/Bs) =

E{E(X(t)/B*)/Bs),

which proves iv). Definition 1.6. The system (dBw(t), Wt'\ F(t, u), b)(t) Y\.; i, j , I = 1, 2, •••) obtained above is called a generalized canonical representation of Y(t). The multiplicity of E(t) is called the multiplicity of Y(t). The classification of generalized canonical representations is very complicated, because, for one thing, the choice of the system {fCD} is not unique. Theorem 1.6. A necessary and sufficient condition that Y(t) has a canonical representation is that Y(t) satisfies the conditions (3JJ. 1), CM. 2) and (9Ji. 3)

The multiplicity of Y(t) is one.

Proof. Necessity. Suppose that (dB(t), SSRt, F(t, u)) is a canonical representation of Y(t) and (1.14)

X{t) = [F(t, u)dB(u) ~ Y{t),

( ~ : equivalent in law)

The separability of 9Jl= \J Ttt is easily deduced from the definition of the integral. Let %Jlt(B) be the same as in Definition I. 4. WltCWt(B)

= l\'f(!i)dB{u);

Then

/ is Borel measurable and

and

r\wit(B) = {0}

eL2(v)\

122

Takeyuki Hida

imply the condition OK. 2). (3JI. 1) follows at once from the separability of \J 2R*Cfl). Finally, if we decompose the random measure B(-) parts in the same way as in (1.1), we can easily see multiplicity of Y{t) is one by Definition I. 6. Sufficiency. From Theorem I. 5 and the assumption multiplicity of Y(t) is one, Y{t) can be expressed in the

into two that the that the form

Y(t) = ['Ftf, u)dBm(u) + 2 t>j(t)Xt.. Now we can define a random measure B{-) by B((a, bl) = B(b)-B(a) = ["dBw{u) + 2 * aJCt,,

{^a)E{XD

< «,).

And define a function F(t, u) of w for every fixed t, by l bj(t)/cij

if u =

tj.

Then we have J V(f, «)rffl(«) = ( V(/, u)dBm{u)+

J}ajF(t, tj)Xt. .

= J Vx(f, u)dBm(u) + j W , «) - i W , u))dBm(u) +

£bj(t)Xtj,

which is equal to Y(t), since the second term of the last expression is zero with probability one. The canonical property of (dB(t), fflt, F{t, u)) follows from iv) in Theorem 1.5. The process Y2{t) which was given in Example 1.2 has no canonical representation, because the multiplicity of Y2{t) is 2, as is easily seen. But a generalized canonical representation exists. In fact it can be expressed in the form Y2(t) = [*F(t, «)rffi1(«)+ ('(1-F(*, u))dBt(u), Jo

Jo

where _ f 1, I 0, Corollary.

if t is rational, if M s irrational.

An additive process has a canonical representation.

Canonical Representations of Gaussian Processes

123

Proof. If B(t) is an additive process, it can be expressed in the form B(.t) =

['dB1{t)+5?Xtr

This shows that the multiplicity is one, and our assertion follows. §1.4.

Kernel criterion for canonical representation.

It is important to give a criterion to determine whether a given representation is canonical or not. P. Levy gave a criterion involving a Hellinger's integral (Levy [6]), but we shall give another. By Theorem I. 2, it is sufficient to give a criterion for a proper canonical representation. Theorem 1.7. A representation (dB(t), Wlt, F(t,u)) is proper canonical if and only if, for any fixed t0 e T and fin Ll{v, U) (1.15)

\*F{t, u)f(u)dv{u) = 0

for every

t
implies (1.16)

f{u) = 0

almost everywhere (v) on (— oo, t0~] f\

T.

Proof. Suppose that the given representation is not proper canonical. Then by (I. 3) there exists an element Z(=4=0) of SSlta{B) which is independent of every X(t), t<.t0. Noting that Z can be expressed in the form Z=[

to

f(u)dB(u),

f e L\v),

we have E(Z-X(t)) = 0,

for every t ^ t0,

which is identical with (I. 15). On the other hand 0 4= E(Z2) = ['0f(u)2dv(u ) . This shows that (1.16) does not hold. Conversely, if there is a function f(u) satisfying (1.15) but not satisfying (1.16) for some t0, then Z = \tof(u)dB(u)

124

Takeyuki Hida

belongs to Tlt(B) but does not belong to 2ft,. Hence the representation is not proper canonical. According to Karhunen [V\, a stationary process which is purely non-deterministic and M2-continuous always has its (moving average) representation, the kernel of which is a function of t~u. He also gave a kernel criterion for canonical (in our terminology) representation. One can easily see that our Theorem 1.7. is a generalization of Karhunen's theorem Example 1. 4. Let Xx{t) and X2{t) be defined by XAt) = \'(2t~-u)dB1(u) Jo

X2(t) = ['(-3t + 4u)dB2(u) Jo

where Bf(t), 2 = 1,2, are ordinary Brownian motions. Then the two processes have the same probability distribution (cf. P. Levy [43), since they have the common covariance function 3ts—2s2/3 {t^>s). Using Theorem 1.7, we can prove that (dB^t), 2t—u) is a proper canonical representation of X^t). On the other hand udB2(u) 0

is independent of every X2{t) {t<.t0), which proves that the representation (dB2(t), — 3t + 4u) of X2{t) is not proper canonical— indeed it is not canonical. Example 1. 5. (Particular case of Example I. 3). If we denote an ordinary Brownian motion by B0{t), X(t) = ['(3-12u/t

+ 10u2/f)dB0(u)

Jo

is again a Brownian motion.

Here

udB0(u) and o

Z2 = I u2dB0(u) Jo

are independent of every X{t) (t^t0). Hence (dB0(t), 3-12u/t 2 + 10u /f) is not proper canonical. In fact, the canonical representation of B0(t) is (dB0(t), 1).

Ill Canonical Representations of Gaussian Processes

125

Section II. Multiple Markov Gaussian Processes. § II. 1. Simple Markov Gaussian Processes. We intend to study multiple Markov Gaussian processes in this section, using the general theory of representation. All the processes to be discussed here are Gaussian processes with mean 0 satisfying the conditions {Sttl. 1) and (2ft. 2). Furthermore we may assume that (9ft. 4)

9JJ, is continuous in t,

that is, lim 9JJ,

exists and is equal to Wt ,

'-*'o

since we can easily remove the discontinuity of Wlt. First we shall treat a simple Markov Gaussian process. Though some of the results are well known, our presentation of the results will stress their specific probabilistic significance form our standpoint. Let Y(t) be a simple Markov process. As Y(t) is Gaussian the simple Markov property is equivalent to the condition that, if s^,t, (II. 1)

E(Y(t)/B,)

=
where cp(t, s) is a real valued ordinary function of (t, s) (Doob [2]). This is also equivalent to (II. 2)

Y(f)-
T<S.

To avoid the case in which Y(t) and Y{s) are independent for s=\=t, let us assume that (II. 3)

V(s, t) = E{Y{t)-Y(s))

never vanishes.

Then the equality E(E(Y(t)/Bs,)/Bs)

= E(Y(t)/Bs),

for every

s^s'^t

implies
'

i
and we can prove that q>{t, s) never vanishes. convention

If we use the

126

Takeyuki Hida
!

f or 5 >

t,


f(t)/f(s),

where f(t) = g>(t, s0) with some fixed s0. Hence we have, from (II. 1), which proves that U(t)=f(ty1Y(t) is an additive process. Here we should note that the system of Borel fields relative to U(t) is the same as that relative to Y{t), since f{t) never vanishes. According to the Corollary to Theorem I. 6, U(t) has a canonical representation, which has no discontinuous part as we assume (Tl. 4). Hence so does Y(t): Y(t) = f(t)U(t) =

f(t)^dU{u).

Conversely, a process expressed in this form is obviously a simple Markov process provided that f(t) never vanishes. Summing up, we have Theorem H. 1. Under the assumption (5K. 1), (9K. 2), (9ft. 4) and (II. 3), a necessry and sufficient condition that Y(t) is a simple Markov process is that it can be expressed in the form (II. 5)

Y(t) = f{t) U(t) = f(t) J 'dUiu) = J 'fit)giu)dBiu),

where Uit) is an additive process with the property (StJi. 4) idBit) is a continuous random measure) and fit) never vanishes. Making use of this theorem, we have (under the same assumptions) Corollary 1. Let Yit) be expressed in the form (II. 5). / / Y{t) is continuous in the mean, then fit) is continuous and Uit) is continuous in the mean. Proof. If Yit) is continuous in the mean, then lim EiYit)Yis))

=

EiYit0)Yis))

by the continuity of inner product in L 2 (0). be written as,

By (II. 5), this can

113 Canonical Representations of Gaussian Processes lim f{t)f{s)E{U(t)U(s)) = f{t0)f{s)E{U(sf),

127

t>s,

since U(t) is additive. Noting that /(s)4=0 and E(U(s)2)=¥0, we can see the continuity of f(t). The continuity of U(t) follows from E(U(t)U(s)) = r(t, s)/(f(t)f(s)). In particular, if T=[0, °o) and E{U{tf) has a continuous derivative, U(t) becomes an ordinary Brownian motion by the change of time scale.13 In other words, Y(t) has a canonical representation (dB0(t), f{t)cr(u)) with Wiener's random measure B0(-) and a proper canonical kernel f(t)a-(u). Corollary 2. / / Y(t) is a stationary simple Markov process satisfying the conditions (Wl. 1), (Wl. 2) and (II. 3), then it has a version (II. 6)

c\' e-Kt-">dB0(u),

\>0.

Proof. As is easily seen in the proof of Theorem II. 1, the covariance function
ry(h) = f(t + h)f{t)
h> 0,

even though we do not assume (W. 4). Putting if=0 in (II. 7), we have f(h) = cl7(h), and putting h = 0 in (II. 7), we have = c37(/?)-2. Hence it follows from (II. 7) that C3ry(h + t) = Ci7(h)-Ci7(t) . Since 7 is bounded above, we have 7(h) = cV x "",

X>0.

§ II. 2. Multiple Markov Gaussian processes. In this article we shall define iV-ple Markov process as a generalization of simple Markov process and study its properties. The property (II. 2) for a simple Markov process suggests that it is natural to give the following Definition n. 1. If {E(Ftf,)/fi, o )}, 1 = 1, 2, •••, N, are linearly 1)

See Seguchi-Ikeda [ 1 ] .

128

Takeyuki Hida

independent for any {t{} with t0-^.t1<^t2<^-"<^tN, and if {E(Y(ti)/ Bto)}, i=l, 2, ••• ,N, N+l, are linearly dependent for any {*,-} with ti
(II. 8)

X{t) = J ' g fMgMdBiu)

with a proper canonical kernel ^fi(t)gi{u),

where {/,•(/)}, «' = 1, 2, •••,

iV, satisfy (II. 9)

det (/,•(*,-)) =4= 0,

for any TV different t, ,

awo" {&•(»)}, i = l, 2, ••• , iV, are linearly independent as the elements of V{v; /)25 for every t. Further the covariance function r of Y(t) can be written in the form r(s,*) = 2/i(f)A,-(s),

s
w/ijere {fi(t)}, 2 = 1,2, ••• , iV, are ^/ze same as a t e a«a* {A,-(s)}, « = 1, 2, ••• , N, are linearly independent. Proof. By the assumptions there exists a proper canonical representation (dB(t), F(t, u)) of Y(t): Y(f)~X(t)

= JV(/, u)dB(u),

( ~ : equivalent in law)

It is sufficient to determine the form of F(t, u) in the region D= {(u t,) ; utN there exist {«/(*"; ti, t2, ••• , /JV)} .7 = 1, 2, •••, TV, such that (II. 10)

Y(r) --Eaj(r;tlttt,-,tN)

Y{t})

/=i

is independent of every Y{
Therefore we have

2) L2(v; t)={ / } .

Canonical Representations of Gaussian Processes F{fr, u){F{r, « ) - 2 af? ;tx,t2,-,

129

tN)F{tp u)}dv{u) = 0 ,

using the representation of Y(t). Since F(
(II. 11)

as an element of U{v; tj (see Theorem I. 7). Take N different {sj} with s1 arbitrarily in the interval (—°°, tt)r\T. Expressing F(r, «) and {F(/y, «)}, j=l, 2, •••, A/", in (II. 11) by {FCs,., u)}, 3=1, 2, ••• , iV, in the same way as in (II. 11), we get N

2 a.(r ; slt s2, ••• , sN)F(s,, u) = 2 « * ( r ; ^i. ^, •••, *JVK(** ; s1; s2, •••, sN)F(s,, U) as an element of U{v; st). Here {F(sf, «)}, j=l, 2, ••• , iV, must be linearly independent functions in U(v; s j ; in fact, if this is not true, then {E(Y(s})/BSl)}, 1=1, 2, ••• , N, are linearly dependent, which contradicts our assumption. Hence we have IT

2 akiT ; *i> t2, ••• , tN)aJ.(tk; slt s2, ••• , sN)

(II. 12)

= aj(r ; slt s2, ••• , sN),

for every j .

Now we can prove (II. 13)

det (cij(tk; sl} s2, ••• , sN)) =# 0 ,

because F(f., u) = 2«*(^-; ^i, 52, ••• , sN)F{sk, u),

j = 1, 2, ••• , N,

are linearly independent functions in L2(y ; s j . Therefore we have (II. 14)

a(r, t) = a[r, s)B(s, f)

by (II. 12) and (II. 13), where a{r, s) = («,(T ;
j , k = 1, 2, ••• , iV,

116 130

Takeyuki Hida

with det (B(t, s))=t=0. Taking TV different ^ « s 1 ) , we have O(T, t) = a(T, *)fl(s, t) = a(r, / ) £ ( « ' , s)5(*, t) , a(T, *) =a(T,J)B(J,t), by (II. 14). Hence we have (II. 15)

Bis', s)B(s, t) = B(s', t).

Fix all t/s and define f,(r),

s = (slt s2, ••• , sN), by

f.[r) = a(r, s)B(s, t)

for

r > s^,

where s is any iV-ple ( v s2, ••• , s^) such that tN'^>tN,1--- ^>^^> SAT^'SAT-I^"- ^> 5 I- Then we can use (11.15) to see that fs> is an extension of fs(r) if sN^>sN-1^>---^>s1^>s'N^>s'Jll_l'^>~-^>s[. Hence there exists a common extension for all / , ( T ) ' S . We denote this common extension with f{t) = {ft{t), ••• , fN{t)). Obviously these fi{t) satisfy (II. 9) on account of (II. 13) and the definition of fB{r). Take u 6 T° and fix it. If T > ^ > ••• > ^ > S J V > ••• > « ! > « , then we have IT

F(T, « ) ( ^ 2 «/(T ; *., *2, •••, tN)F(tj, u)) = «(T, *)F(*, «)* (F(*. ») = (F(tlt u), ••• , f(f„, «)) _1 = / ( T ) B ( S , *) F(s, M)* = f(T)tf(«, s, *)*,

(£(«, s, t) =F(s, u)B(s,

For T^>tN^>---^>t1^>s'N---^>s[,

t)*-1).

this is equal to

f{r)g(u,s',

t)* ,

so that f(r)g(u, s, t)* = f(r)g(u,s', for r > m a x (s'N,sN).

t)*

Since f satisfies (II. 9), we have

g(u, s, t) = g(u, s\ f). Therefore g{u)=g(u, s, t) is well defined as a function of u, and F(t, u) = f(f)g{u)* = 2/,(*)&•(«)> where {|f,(«)}, « = 1, 2, ••• , iV, are linearly independent as elements of L2iv ; t), since {Fitj, u)}, j=l, 2, ••• , N, are linearly independent. Further we have

117

Canonical Representations of Gaussian Processes nt, s) =

131

^f,{f)(i:fj{s)['gMgj(u)dv(u)) ,•=1

;=1

J

Then if

f] aMs) = 0 ;=i

for some constants ax, a2, ••• ,

aN,

[Sdfj(s)gj(u))(J2 J

Noting that ^/As)gAu)

aigi(u))dv(u) ^ 0 . i=l

;=1

is a proper canonical kernel, we have

,=i

2 «,-&(«) = 0 . ;=i

Hence all the a, must be 0. completely.

Thus we have proved the theorem

A kernel ^2fi(t)gi(u) satisfying the conditions stated in Theorem <=i

II. 2 is called a Goursat kernel of order N. It should be noted that the expression (II. 13) is not uniquely determined, but the number of the summand is independent of the special way of expression as we have seen in the proof above. As another remark, we should note that a process with a version of the form (II. 9) is not always an iV-ple Markov process. In order that the converse of this theorem holds, it is sufficient to impose some regularity condition on the kernel as we shall see in §11.4. 3°). Example II. 1. If f(t) is a function which is 1 for rational t and 0 for irrational t, then X(t)=f(t)B0(t), 0
X{t) = ( 7 ( W ( « ) • Jo

§ II. 3. Stationary multiple Markov Gaussian processes. Let Y(t), /eT=(—oo,oo), be a stationary Gaussian process with mean 0 satisfying the conditions (W. 1), (2Ji. 2) and (9Ji. 4'). (Sffi. 4')

Y(t) is continuous in the mean.

132

Takeyuki Hida

Then by Karhunen's theory, we can see that Y(t) has a canonical representation and that it is expressed in the form Y(t) = j ' F(t-u)dBt{u),

E{dBa(u)f = du ,

using a canonical kernel F(t—u). This canonical kernel is uniquely determined up to sign and is proper canonical.33 Thus, in order to study the multiple Markov process for stationary case, it is sufficient for us to study the canonical kernel F(t—u). Lemma II. 1. Let {/„•(/}}, i=l, 2, ••• , N, satisfy (II. 9) and let {gi(u}}, i = l, 2, ••• , N, be linearly independent as elements of U{{~ °°, c~\) for every c. If ^fi{t)gi(u)

is a function of t — u in

the domain D= {{u, t) ; u^.t}, then {fi(t)} is a fundamental system of solutions of a certain linear differential equation of order N with constant coefficients, and {&•(«)} is also a fundamental system of solutions of its adjoint differential equation. N

Proof. First we consider F{t—u) = '^ifi{t)gi{u)

in the region

i=i

Da= {(«, t); u<:0, t^O}. Let ®0 be the set of all C°° functions whose carriers are compact sets lying in the interval (— » , 0]. Then (F*
F(t-u)q}(u)du

is well defined by the assumption and belongs to C°°((0, <»)) for every
det ((£,•, cp.)) 4= 0,

i, j = 1, 2,... , N,

where (g, 0 such that (g1, ,)) 4=0,

i,j=l,2,-,n.

And consider the determinant 3) For proof see Karhunen [1]. Also, M. Nisio gave another proof, which I knew by private communication.

Canonical Representations of Gaussian Processes (gi,


(gi,
(g„
(gi,
(gi, 9>l)

(gi,
(gi,


(g«+i,
(gn+K^l)-"

(gn+i,
133

(g«+i,
If this determinant vanishes for every

-0,


by expanding this determinant with respect to the last column ; since

A

- A »+I£»+I(W) = 0

2&(«)-

a.e.

in

(-

',0)

This contradicts the assumption that {gt(u)}, i = l, 2, ••• , « + l, are linearly independent, since AM+1^=0 by the assumption of induction. Thus we can take {
S

,)(*) = 2 (&,?>/)/.•(*)

and (II. 16), we can see that ft(t) is a linear combination of (F*
1 = 1 , 2 , " . , TV.

Thus we have

(II. 17)

g/?>(*)&(«) = J^i'-") = t-D*^*^-") (-i)*2/*(*tei"(«),

& = 0, 1, ••• , N.

Putting w = 0, we get ^Fit)

= (-l)*g/,•(*)*<,"(()),

* = 0, 1, ..., N. jsr

Therefore there exist b0, blt ••• , bN such the XII&.I^X) and that b0F^(t) + b1F^-1\t)+so that

+ bNF(t) = 0,

*>0,

134

Takeyuki Hida b0F^N\t~u)

+ b1F^1\t-u)+-

+ bNF(f-u)

= 0,

/>«,

+ bNfi(t))gi(u) = 0,

t>u.

namely 2 (hA'n(t) + b1ff-»(t)+i= l

Since {g,(w)} are linearly independent in U{{— oo, /]), bJf\t)

+ bJf-»{t)+-

+ bNfi{t) = 0,

i = 1, 2, - , AT.

If b0=0, then /,•(£) satisfies Ci/i(fl + c2f2(t) + •••+ cMt)

= 0.

2 1 c, 14= 0 ,

as a system of iV solutions of linear differential equation of at most order N—l, which contradicts (II. 9). Hence {/,-(*)} is a fundamental system of solutions of linear differential equation of order N with constant coefficients. Exactly in the same way, we can prove the assertion for {&•(«)}•

By the well-known fact in the theory of linear ordinary differential equations, F(t—u) is a linear combination of the functions of the following types: g-*"--> sin pit-it), tku"-ke-K'-u> sin p(t-u), (11.18) *-*"-"5 cos/*{*-«), f V - V * " - " 5 cos p(t-u),

(/*H=0) (ft may be 0)

0
n<:N.

By Theorem II. 2 and Lemma II. 1 we have Theorem n. 3. / / Y(t) is a stationary N-ple Markov process satisfying the conditions (2Jt. 1), (2Ji. 2) and (9ft. 4'), £/ze« «Ys canonical kernel is a linear combination of the functions described in (II. 18) with X^>0. The functions in (II. 18) (for /*=4=0) are split into two terms of the form f(t)g(u). Therefore the number of the terms in the expression of the kernel is exactly N. Corollary. The spectral measure of a stationary N-ple Markov process is absolutely continuous with a density function of the following type: \Q{iX)/P(iX)\\ where P is a polynomial of degree N and Q is also a polynomial of degree at most N—l.

Canonical Representations of Gaussian Processes

135

Proof. The spectral density function is obtained by Fourier transform of F(-). Hence our assertion is obvious. This process is a component process of an iV-dimensional stationary simple markov process in Doob's sense, Doob [1]. §11.4. Some special multiple Markov Gaussian processes. 1°) Let Y(t) be a stationary N-ple Markov Gaussian process which is differentiable (with respect to Z,2(0)-norm) up to N— 1 times. Such process plays an important role in the study of TV-pie Markov processes as is seen in Doob's work [1]. Now let us assume that Y(t) is expressed in the form Y(t)~X{t)

= ('

F(t~u)dB0(u)

with a proper canonical kernel

Then we have the following Theorem II. 4. Let X(t) be a stationary N-ple Markov process. Then i) a necessary and sufficient condition that X(t) is differentiable is F(0) = 0, ii) in this case, there exists a complex number X such that ext4,e-xtX{t) dt exists and it is a stationary (N—l)-ple Markov process. Proof, i) If /z>0,

(II. 19)

~(X(t + h)-X(t))

= ~^+HF{t + h-u)dB0{u) +

7 r f iF(t +

h-u)-F(t-u)}dB0(u).

Since F(t—u) is analytic in D, the first term of the right hand side tends to 0 (in the mean) as h tends to 0 under the assumption F(0) = 0. Hence lim h-\X{t + h) - X{t)) exists. Similarly lim 4-»0-

H i t

h-\X(t+h)-X{t))

exists and

(II. 20)

X'(t) = j '

^F(t-u)dB0(u).

136

Takeyuki Hida Conversely, if X{t) is differentiable, then dX{t) = F(0)dBJit) + dt\!

J-co at

~F(t-u)dB0(u)

will be of order dt, so that F(0) should vanish. ii) As we have seen in Theorem II. 3, f{(t) is a solution of a linear differential equation with constant coefficients. If we choose one of the characteristic roots of the differential equation, say X, ek'^-e-xtF(t~u) is obviously a proper canonical Goursat kernel of order N—l. The existence of (II. 19) and the stationary property are obvious. When A, is real (II. 19) is real valued process. When X is complex, say X=A,1 + A,2, (11.19) is complex valued process, but Ad) -dtfS) -'Xd),

A{t) = eV cos V ,

is a real valued stationary process. If F(t) satisfies the conditions F(0) = F'(0) = - = FiN~^(0) = 0, X(t) is differentiable N—l times. Then we can take a sequence of complex numbers \ , \ , ••• , XN_1 such that eh* 9^«.--.-V ... | e t t r ^ ' | e - \ ' X { t ) = X™(t) dt dt dt exists and it is a stationary (N— i)-ple Markov process. Such a process was studied by Doob [1] and the formula (II. 21) suggests more general differential operator which will appear in 3°) 2°) We shall now discuss a multiple Markov process with a homogeneous canonical kernel; F(t, u) is called to be homogeneous function of degree a if F{ct, cu) = caF[t, u). This process can be transformed into a stationary process by time change by virtue of the following Lemma II. 2. (P. Levy) Let X(t) be expressed as (II. 21)

(11.22)

X(t) = [tF(t,u)dB0(u) Jo

with a proper canonical homogeneous [of degree a) kernel F(t, u).

Canonical Representations of Gaussian Processes

137

Then t~a~1/2X(t) is a stationary process of log t (P. Levy [4 ; p. 141]). Applying this lemma to TV-pie Markov process, we get Theorem II. 5. Let X(t) be an N-ple Markov process and be expressed as in Lemma II. 2. Then e~i2a+l:>tX(e2t) is a stationary Nple Markov process. Conversely any stationary N-ple Markov process X(t) is an N-ple Markov process with homogeneous kernel of degree 0 changing the time parameter from t to e': \/TX{{\ogt)l2)

= X{t).

Proof. The first part of the theorem is an immediate consequence of Levy's lemma, if we notice that TV-pie Markov property is invariant under such time change. If X(t) is a stationary iV-ple Markov process, it is the sum of the processes of the following types: (11.23)

(' e'w'"^dBa{u),

\' {t~u)keKa^dB^u)

.

Changing the time scale, they become

(ii. 23') ^=\\u/ty^dB 0 (u), (^=)* +1 JW (u/t)nu/ty^dB0(U) respectively. Hence X(t) is an iV-ple Markov process with homogeneous kernel of degree 0. If X is complex, these expressions are not real, but we can reduce them to real ones by the same procedure as used in the proof of Theorem II. 4. i). 3°) Let us generalize the results obtained in 1°) and 2°) to the case in which X(t) is a general N-ple Markov process: our results include also those which were discussed by Dolph-Woodbury [X\ and Levy £4]. For the sake of brevity we shall assume T=[0, oo). We can discuss stationary iV-ple Markov processes with parameter €(— °°, oo) in this scheme, if we apply to it the time change used in 2°). We shall consider a process X(t) which is expressed in the form (II. 24)

X(t) = T E fi(t)gi (u)dB0(u), Jo i=l

with a proper canonical Goursat kernel 2 / . (*)&•(")• Hereafter (throughout this article), we shall always impose the following conditions on the kernel:

138

Takeyuki Hida

(A. 1) (A. 2)

/,-, gi 6 C"(T°),

for every i,

ft and W(glt g2, ••• , gi) never vanish for every i,

where W(glt g2, ••• , gt) is the Wronskian of {g3), j=l,2, — , i. By these assumptions, we can find functions v„,vlt---, vN_ such that (II.25)

gi(u)

= (-l)N-iv0(u)\av1(u1)[\(u2){"2 Jo

-

Jo

J

Jo ir-i-l

M

Vx.iiuN-iHdu)"-'

0

and that (II. 26)

f

vM Vi

e CT(T°), (u) never vanishes,

i = 0, 1, • • • , A7— 1.

Using these functions {«,(«)}, i=0, 1, ••• , A7—1, and a function vN{u) satisfying (II. 26), we can define measures ntiiM) = \ Vi{u)du, MeBT,

i = 0, 1, ••• , N,

and the following differential operators: '

dm0dm1 dmN_t vN{t) ' d d d 1 T U) dmJ-dmJ+1 dmN_1 vN{t) L* = the adjoint operator of Lt _ d d _ _d_ m _ ^ _ m dmNdmN_1 dm^ v0(u) d d d 1 dmN_jdniN_J_1 dm1 v0(u) . We shall often use the following notations: F(t,u) = J}fi(t)gi(u), F^(t, u) = ^.F(t,

u
«),

FWtf, w) = L?-»F(t,

u).

Theorem II. 6. Let X{t) be defined by (II. 24) «wJ /rf £fe canonical kernel of its representation satisfies the conditions (A.1) and (A.2). / /

Canonical Representations of Gaussian Processes (II. 27)

139

F(t, t) = Fm(t, t) == • • • FCN~2'(t, f) == 0 and F'N^(t, never vanishes,

t)

then we have i) XCD{t) = jnXit)

exists for every

i
ii) X(t) satisfies the equation (II. 28)

LtX{t) =

B'S),

where B'0(t) is a derivative of B0(t) in the symbolic sense, so that (II. 28) means dL^X{t) = v0(t)dB0(t), and the measure mN associated witn Lt should be taken appropriately. Proof, i) is proved in the same way as in the stationary case (Theorem II. 4, i)). ii) Define vN{t)=f1(t). Since vN{tYKF(t,t)=Q by assumption, we can prove the existence of -^v^fY^Xit); namely Lf~^X(t) exists. Similarly, we can prove the existence of L^X(t), i = N—2, N-3, ••• , 1, since F™(t, t)=F^(t, t)=-=F^-2Ht, f)=0. Rewriting (II. 27) in the following forms 2/$"(«&(*) = 0,

k = 0, 1, - ,

N-l,

.•=i

flff-^WgAt)

= a(t),

with a(t)=F
we can see that F(t, u)/a(u) is a Riemann function for a certain linear differential equation L,/=0 of order N. The fundamental system of solutions of its adjoint differential equation L*g=0 is {gi(u)Ia{u)}, i = l, 2, •••, N, as is well known. Hence L* = L*-v(u) with a certain function v{u). Thus we can prove that Lt = v(t)-Lt. By the property of Riemann function v(t) must be 1, and therefore we have (II. 29)

/,-(*) = vN(t)\ dm^

jdm,v- 2 J ••• jdmJV_,+1, i=l,2,-,

iV,

140

Takeyuki Hida

which proves that L^F{.t,u) = gN{u) = v0(u). Hence V?X{t) = [L^Fit,

u)dB0(u) = [v0(u)dB0(u).

Jo

Jo

Thus we can prove ii). Combining this Theorem II. 6 with Theorem II. 9. in § II. 5 we can see that this X(t) process is an A'-ple Markov process in the restricted sense in Levy's terminology Corollary. Under the same assumptions as in Theorem II. 6, Lt"X(t) is an (N—i)-ple Markov process in the restricted sense. Proof. As is easily seen in the proof of the theorem above, D?X(t) = [L^Fit,

u)dB0(u).

Jo

Since L\^F{t, u) is a Riemann function for the differential equation d ... d r _ p. dmt dmi_t '

our assertion is obvious. Theorem II. 7. Let {«,-(«)}, * = 0, 1, ••• , N, be functions satisfying the condition (II. 26). / / we define fi{t)'s and gi(u)'s by (II. 29) and (II. 25) respectively, then IT

i) F(t, u)= ^2fi{t)gi(u) is a proper canonical kernel, ii) a process defined by X(t) = ['

J*Mt)giMdBt(.u)

Jo i = l

is an N-ple Markov process, iii) F(t, t) = Fm(t, *) = . • • = F™-*>(jt, /) = 0 and FCN~^(t, t) never vanishes; namely Theorem II. 6 holds for this process. Proof, i). We shall prove i) by using the kernel criterion which was given in § 1.4. Suppose that I F(t, u)
in (0, t0)

Jo

for some t0 G T and some

J0

Writing it in the form

Canonical Representations of Gaussian Processes and multiplying vN{ty\ Nikodym sense)

141

we can obtain its derivative (in Radon-

JlA11(t)\tgi(uMu)du + (tfi(t)gi(t))
(=1

J 0

a.e. in (0, t0). Since the second term of the left hand side vanishes (because ir

Hlfi{t)gi{u) is a Riemann function corresponding to Lt), we have ;=i

2/ ( "(t)['gi(uMu)du .' = 1

= 0,

in (0, t0),

J0

by taking an appropriate version. Repeating such procedures, we can prove vtf) \ v0{u)q>{u)du = 0,

in (0, t0) .

Jo

Hence
A(tly t„ ••• , tN) = d e t (/At,)) 4= 0 for any different

t/s.

If A(^1; t2, ••• , tN) = 0 for some t1<^t2 we can prove D(tltt2,-,

tN)

\ vN^xdu,

\ vN_xdu,

since i v never vanishes. VN-At'l), »JV-IW)\

Jo

vN.2du,

1

vN_1du

By the mean value theorem, we have

fliV-iW), 1

•••, I

•",»iV-i(^-i)

vN_1(t'2)\vN^2du,---,vN^(t'jt^l)\ Jo

^vN_2du Jo

•••,vN^{t'N-^"-\vx{du)N-z

142

Takeyuki

Hida

for some {/•}, i = l, 2, ••• , N—l, with t'( £ {t{, ti+1). vaishes, it is proved that D(t[,t'z,-,

Since vN_x never

^_1) = 0.

Successively we have D(K, t;, - , &_,) = ... = DU\*-*>, t(2*-2>) = o for some {fS"}'s with «*> 6 (/S*"15, f&l"). Finally we have

which contradicts the assumption (II. 26). Therefore, by (11.30), we can find functions {aft; tN)}, j = l,2, ••• , N, such that (II. 31)

tltt2,---,

2 «y(f ; * „ / „ - , *„)/,(/,) = /,(*)

holds for every /. Hence X{t)~j^aj{t;

tlttt,-,

tN)X(tj)

is independent of every X(T) ; r^,t1. On the other hand [E(X{t3) I Bto)}, j=l, 2, ••• , A'", is linearly independent for any choice of t/s with ta<,t1<^---<^tN, since gi(u)'s are linearly independent as elements of L2([0, £„]). Thus the assertion ii) is proved. iii) is easily proved, noting that F(t, u) is the Riemann function corresponding to Lt. Theorem n. 8. Let X(t) be defined by (II. 24) with a canonical kernel F(t, u)=^fi(t)gi{u),

and let fi{t)'s

conditions (A. 1) and (A. 2). If F(t, t)=Fm(t, t) = -=Fik-v(.t,

and gi(u)'s satisfy the

t) = 0 and F^(t, f) =£0

for some k«^N—l) independent of t, then there exist Y{t) which is an N-ple Markov process in the restricted sense and (N—l)-th order differential operator Mt such that (II. 32) X(t) = MtY{t). Proof. By assumptions (A. 1) and (A.2), gi(u)'s can be expressed

Canonical Representations of Gaussian Processes

143

in the form (II. 25). Therefore there exist a differential operator &~ Lt and a Riemann function R(t, «) = 2/«(*)&•(«) corresponding to Lt, where /,(/) is defined by fi(t) = vN{t)\ dmN_\dmN_\--AdmN-i+1,

i = 1, 2, •••,

N.

Define Y(t) by Y(t) = [*R(t, u)dB0(u). Jo

Then, this Y(t) will be the bne to be obtained. The assumption (A. 2) implies W(flt f2, •••, 7^)4=0 for every t. Therefore we can find a differential operator Mt such that MJAt) = Y.btfWit)

= /,(/),

i = 1, 2, - , N.

Noting that Y(t) has the j-th order derivative Y^(f)

= ['R^V,

u)dB0(u)

Jo

for every j^N—1, and we have

by Theorem II. 6, i), Mt can be operated to Y{t)

MtY(t)

^JlbjWYnt) pf j r - i

=

[j]bJ.(t)R^(t,u)dB0(u) Jo y=i

=

[ixilbjWfiltngMdBM J o 1=1y=i J 0 1=1

This completes the proof. Example II. 2. Levy's example Xtf) which we discussed in Example 1.5 satisfies all assumptions imposed on the canonical kernel in Theorem II. 8, where N=2, v0=v1 = l and fe=0. In this case Y(t) = andM,= ^ .

[\t-u)dB1(.u)

144

Takeyuki Hida

§ II. 5. Prediction of multiple Markov Gaussian processes. For a Gaussian process Y(t), the least square linear prediction on the basis of its values before s(
E(X(t)/Bs) = \SF(t, u)dB{u),

(X(t) = j*F(/, u)dB(u)).

It is our aim to express it in terms of X(r), rf^s. Theorem n. 9. Let Y(t) be a process defined in Theorem II. 8. Using the same notations, we have (II. 34)

E(Y(t)/Bs)

= f ] btftfYV-Vis),

s
where

(11.35)

bJ(t,s) = J2f,{t)±,t/Ms)

with A(s) = det {f\t~^{s)) and A;>. = (;, i)-cofactor of A(s). Proof. Putting U{(s)= \ g;(u)dB0(u), we have Jo

(II. 36)

E(Y(t)/Bs) = (*£i/i(f}&(«)<*3>(«) = Ilfi(t)U,(s) • J 0 1=1

1=1

On the other hand, Y™(s) = L^-"F(5) =

fj

/M(s)f/,-(5),

* = 0, 1, - , i V - 1 .

i=*+ l

Since A(5) = (/f«(5))

=£[^(5)4=0, 1 = 1,2,-".Jf 4 = 0 , 1 , ••-,JV r -l

i=l

£/,(s) can be written in terms of F m (s)'s, k = 0, 1, ••• , N—l, that is

Us) f2(s) (11.37)

t/,(5)

A (5)

- Y'(s)

... /„(*)

o fp(s) - yw(5)

- /m(5)

0

0

-

••• f%Hs)

o

o

- y^-'](s) ••• fif-ms) .

Combinig this with (II. 36), we have

Yra(s)

Canonical Representations of Gaussian Processes E(Y(t)/Bs)

=

145

ibfi(t)(ib£±YV->Hs))

1=1

\/=iA(s)

i=i\.=i

/

A(5)/

which was to be proved. Corollary. Let X(t) and Y(t) be the same processes as in Theorem II. 8. Then we have (II. 38)

E(X(t)/Bs) = 2

Cj{t,

s)YU-*(s),

i=i

where A(s) A(s) and A.^ being the same as in (II. 35). Proof. Noting that E(X(t)/Bs) = 2 fid)U,(s)

for every

s
.=1

we can easily prove (II. 38). This corollary suggests the following symbolic calculous of determing the predictor. Using the differential operator Mt defined in (11.33), (11.38) becomes E(MtY(t)/Bs)

= itcj(t,

s)YV-V(s)

y=i

= SM^W^-1^) = This means that Mt and E(-/Bs)

MtE(Y(t)/Bs).

are commutative.

On the other hand, (II. 38) may be denoted as E(X(t)/Bs) = 2

Cj(t,

s)(M^X(t))li-^

,

y=i

where Mjl is an integral operator such as Mt{MjlX{t)) = X(t). Hence formally speaking, the prediction operator for X(t) is composed of differential and integral operators. Theorem n. 10. Under the same assumption as in Theorem II. 9, N

lim 2 a At, slts2,--,

sN)Y(s{)

146

Takeyuki Hida

exists and equals E(Y(t)/Bs), where a{{t, s„ s2, ••• , sN), i = l, 2, ••• , N are the functions determined by Theorem II. 7, (II. 31). Proof. Refering to the proof of Theorem II. 7, we have i W , slf s2, - , sN)Y(Si) = 'ht^r-j--

rS

fywW'>

where A^fo, s^, ••• , 5^) has been defined in Lemma II. 5 and A^f> is its (j, i)-cofactor. Letting s1} s2, •••, 5^ tend to 5 successively, N

we can easily prove that 2#«(*> slt s2, ••• , sN)Y(si) tends to .=1

i = 1/S(S)

;=1

i= l

as was to be proved. For stationry case, such prediction problem is well known (cf. J.L. Doob [1], [2]) § II. 6. Sum of stationary multiple Markov Gaussian processes. As we discussed in § II. 3, any stationary iV-ple Markov process is considered as the sum of stationary iV,—pie Markov processes in the restricted sense with XI Ni = N. The converse problem will be discussed here. For the sake of simplicity we shall consider the sum of stationary simple Markov processes, which is 1-ple Markov process in the restricted sense. General cases are treated similarly. Let Yj(t), j=l, 2, ••• , iV, be stationary simple Markov processes. Taking appropriate versions we can express Yj{t) with respect to the same random measure B 0 (0 as follows (11.39)

Y,(t) = A*

«-x/'-"'rffl0(«),

X y (>0), c}\ constants j = 1, 2, - , N.

Now let us consider (II. 40)

Y(t) = 2 Yj(t) = (' 2 Cje-W-'>dB0(u)

Obviously it is at most N-ple Markov stationary process. Even in the cast that all the X/s are distinct, the kernel 3

F(t—«)=Scje"¥'"") .-=1

J

Canonical Representations of Gaussian Processes

147

is not always canonical (Example II. 3), and Y(t) is not always Nple Markov process (Example III. 3.). Let F(X) be the Fourier transform of F(x); /"(\) = —L=C e-*\ VZTTJO

F(x)dx.

Then /(X)

(II 41)

= ^MX1~

,

(*' =

V^l),

where Q(iX) is the polynomial of iX at most of degree TV—1. Writing Q(iX) = 21«y(*>-)w"1"-',

(II. 42)

we have Theorem n . l l . F(t—u) is the proper canonical kernel if and only if Q{x) has no zero point with positive real part. Proof. We use the kernel criterion proved in § 1.4. Define the numbers #v, v = 0, 1, ••• , N, by

n(*+\) = 2 ^ - ' . y=i

v=o

And define the differential operator L, by N-1>

L

Y

,dt) ' = MT

Then we can easily see that Lte~xf*'u^ = 0, consequently

L,F{u-t)

= 0.

Now suppose (II. 43)

J' i t y - u)
where ® = {
F{0)(t-u)
k = 0, 1, •••, JV-1

Then from (II. 43) and (II. 44), we have

m(^bN_k^\t))+F>mi}bN-k^h-1\t))+ 4=0

+ Fr-'MCOZypW + j '

•••

i=i

LtF(t-u)
148

Takeyuki Hida

that is,

aV^oHsX-*-^*-"**)) = oIf we introduce a new differential operator

' § a\dt)

L =

'

j

with «,- = S Fm(0)b,v, (II. 45)

then the above equality can be written as L,9>(/) = 0 .

Non trivial function

"§ Six"-1-' = 0

of Lt has at least one root with positive real part. On the other hand, noting that f e-ix*FM{x)dx

= Fc*-15(0) + (/X)Fc*-2)(0) + (a)2Fc*-35(0)+ ••• + (iX,)*" 1 ^)

and F(\) f; bMX)N'j = QOV , we can prove aj = aj

for every j ,

Hence the desired condition is equivalent to the one that

(II. 46')

"2 a,xN'^ = Q(x) = 0 .

has no root with positive real part. Generally, not assuming that

Canonical Representations of Gaussian Processes

149

Af-ple Markov process in the restricted sense if and only if all the a/s are zero except a0. Example II. 3. Consider a process X(t) = 3\' e-«-^dB0{u)-4^

e^'-^dB^u).

The kernel 3e~c'"M)—4e~2C'^K:I is not a proper canonical kernel. Sction III. Levy's M(t) process. § III. 1. Definition and known results. Let X{A, a>), AeEN (iV-dimensional Euclidean space), oaeO, be a Brownian motion with a parameter space EN, that is / (III. 1)

i) X(A) is a Gaussian random variable with mean 0 for every Af

) X(O) = 0, where O is the origin of EN, iii) E(X(A)-X(B))2=r(A, B), where r(A, B) denotes the I, distance between A and B.

I

u

Since X(A, a>) is continuous in A for almost all a>, (P. Levy [X\, T. Sirao [1]), the following integral is well defined and we have a Gaussian process MN(t) with a parameter space T = [ 0 , °°), (III. 2)

ikW*) = (

X{A)do-(A),

J SCO

where SN(t) is the sphere in EN with radius t and da- is the uniform measure on SN(t) with a-{SN{t)) = 1. P. Levy studied the canonical representation and the Markov property of this process when N is odd (P. Levy [3], [4]). Since E(MN(t)) = 0, the covariance function of MN(t) is (III. 3)

VN(t, s) = E{MN(t)MN(s))

\ J

E(X(A)X(B))da-(A)da-(B)

= JA£S„ct)JI. (t + s-PN(t,s))/2 where M ' , s) = [

[

r(A, B)do-{A)do-{B).

By the simple computations we have, for t = s, (III. 4)

PN(t,

t) = tJN_JIN_2

(N^ 3)

150

Takeyuki Hida sin*^fi?6»and A = \ sin kQ sin -^dG, and for s=\=t

with Ik=\ Jo

(III. 5)

Jo

/

2

pN(t, s) =

1 [% sin"" 0 d6

(N^ 3)

with r=tf 2 +s 2 -2tecos6>) 1/2 . Using the analytic property of TN(t, s) and others, P. Levy [4] obtained many important results concerning MN(t). First, if N=2p+1, MN{t) may be expressed as (III. 6)

MN{t) =

VpN(ult)dB0{u), Jo

where PN( — J is a canonical kernal defined by PN{u) = - ? L V j ^ j j l - * 2 K " W *

(III. 7)

= polynomial of degree 2/>—1. For example Example III. 1. \t(2/3-u/t+u3/3t3)^^'dB0(u)

M5{t) = Jo

Example III. 2. ATT(f) = ('(2/5-3«/4if +

3 M

/2it 3 -3M720f)v / i()^5 0 ( M ).

Jo

Concerning the Markov property, it was proved that M2P+1(t) has continuous derivatives of orders l,2,---,p and it is a (£ + l)-ple Markov process in the restricted sense. § III. 2. Canonical representation of MN{t) process. We are now interested in the canonical representation of MN(t) for the case that N is even particularly. First we shall consider some properties of ^N(t, s) for odd and even N, and then we shall study MN{t) process. As P. Levy pointed out e~'MN(e2t) which will be denoted by XN{t), becomes a stationary Gaussian process with parameter space (— oo, oo). In fact (III. 8)

E(XN(t)XN(t + h))= 1- (2 cosh h - cos0Y/2smN~2Gdo)

-i

^ J*(cosh (2h) h^O.

Canonical Representations of Gaussian Processes

151

It is a function of h and will be denoted by yN(h). Lemma. If N^4, 7N(h) belongs to C2 and satisfies the following equation (III. 9)

(2N-3y7N(h)-7Z(h)

=

4(N-l)(N-2)7N_2(h).

Proof. If we note only the differentiability under the integral sign in the formula (III. 8), we can easily prove the existence of y'Ah) and y£(h). Exact forms of them are YAh) = sinh h - ^ ^ L f * {2(cosh 2h - cos 0)} ~1/2 sin^'fl d0 7"N{h)

= cosh h- - , 1 | . — (* {2 cosh 2/*(cosh 2h - cos 6) - cosh2 2h +1} {cosh 2h - cos 0} "3/2 sin""2*? d0

Thus we obtain (III. 9). Theorem i n . 1. / / N^4, (III. 10)

we have

cNXN_2{t) = e-™->'4-*™-»XM(t),

cN =

2V(N~D(N-2).

(Here a process and its version are identified) Proof. From the above lemma, we can see that ei2N~3:,tXN(t) is differentiable. On the other hand, XN(t) is purely non-deterministic as is easily seen from the definition, and it is expressed as XN(t) = \eitxdZN(X) with a Gaussian random measure ZN('). X^(t)

=

Hence

e-cm-™^-[eCik+2N-™dZN(X)

exists and is UiX + 2N-3)eitxdZN(X)

.

The covariance function of Xm(t) is je<**{\2+ (2N- 3)2} | PN(X) 12dX = -vm

+ (2N-3)27N(h)

=

cllN_2{h),

where \FN(X)\2 is the spectral density function of XN(t). X&(t) can be regarded as a version of cNXN_z(t).

Hence

152

Takeyuki Hida

Thus we can see by (III. 9) and (III, 10) that the study of XN(t) is reduced to the study of X2(t) or X3(t) so far as we consider. More exactly, if we denote the operator C^e'^'™-rfei2N~3yt

by DN,

it is easily proved that D3 can be operated to X3(t) and D3D5-D2p+1X2p+1(t)

=

Xl(t),

which is the Ornstein-Uhlenbeck's Brownian motion. Hence X2p+l(t) and therefore M2p+1(t) is a (p + l)-p\e Markov process in the restricted sense. (This fact was proved by Levy by another method). The spectral distribution function of X2p+1(t) has a density p

(111.11)

i^

2

CXI 1 =

1 2p+iy n

*~i 1 {{4p-l)2 + X2} {(Ap-5)2+X2} - {32+X2} 1 + X2

Therefore we obtain the following

X2p+l{t) = j ' {ha^-w^

+

a^'-^dB^u)

Here the kernel of the representation is to be determined so that the square of its Fourier transform is equal to | F2p+1(X) | 2 and it satisfies the condition that Q(X) = constant. This is posible. Changing the time scale, we have the canonical representation of MN(t). J o v 2 *=i

Obviously thus obtained representation coincides with the Levy's result. The problem to obtain all non canonical representations of M2fi+1(t), where kernels are polynomials of (u/t) of degree 2p — 1 is easily solved, if we observe the spectral density function of X2p+1{t). The answer of the problem is that "the number of different representations of above stated form of M2p+1(t) is just 2P~X including canonical one". Proof. If a kernel is a polynomial of (u/t) of degree 2p — l, it turns into the sum of exponential functions such as e-^+1^t-^! k^,2p — l, the Fourier transform of which is a sum of the functions of the form ^-—7?ii_—^. Hence the number of posible functions 2A.+ (Zk+ 1) of ft2p+1(X) is just 2P~1. For example we obtain such a function

139 Canonical Representations of Gaussian Processes

153

multiplying \ ^ , . ~ ; to the function fi2l>+1(X) which corresponds iX+(4p—7) to the canonical kernel. Example III. 3. One of the non-canonical representation of M7(t) different from the Levy's one (cf. P. Levy [4] p. 146) is given as follows: let p<\\=/ c \ & —5 n ; \(iX + l)(iX+3)(iX+7)(i\+U)) ' iX+5 The rational function in the bracket corresponds to the canonical kernel. Then M7(t) is expressed as \t(3/5-3u/t

+ 5u2/t2-3u,>ltd +

2u5/5f)^'10dB0(u).

If N is even, we also have DtD6-D2PX2fi(t)^X2(t). Hence, if we know the canonical kernel of the representation of X2{t), then we can obtain that of X2p(t) easily and know the properties of it. We have p

( I I I . 12)

,fi

/-> \ I 2 _

' ^

"

k-1

{X2+(4i>-3)2}{X2+(4/>-7)2} ••• {\2+52} '

|PA)|2 A

,l

in the way similar to the case in which N is odd. Now it is our purpose to obtain the exact form of \ft2(X)\\ To do so, let us consider y2(h). If h^>0, y2{h) = cosh h-^-

(*(2 cosh (2h) - 2 cos 6fHQ

Using the Legendre's polynomials the integral term of it may be expanded as follows L - H c o s h (2/^cos0} v W0 =4-{«*+ f 3 « i + «*-2a*«Jk+l)e-c4*+"*} , V2 where _ 1.3-5 — (2fe-l) 2-4-6 -2k.

«* =

Taking the Fourier transform, we have

lAMI^ifT-^r.-S 2^V1 + X2 f^>X2+(4k + 3y

154

Takeyuki Hida

where bh = (4k+S)(ak+1-akr

<>0).

This proves that X2{t) is not a multiple Markov process, but, as it were, °°-ple Markov process. Let FC"3(X) be given by

Then i ^ ( \ ) > 0 (since ^ w M > | £ ( X ) | 2 ^ 0 ) and

Therefore, there exists a stationary Gaussian process X^it) is expressed in the form

*<">(*)= J'

which

F„(t-u)dB0(u)

and has spectral density function Fw(\). w Obviously X (t) is a stationary («+l)-ple Markov process and its covariance function yw(h) converges to
Canonical Representations of Gaussian Processes E. L. Ince K. It6 (1944).

155

£ 1 ] : Ordinary differential equations. Dover Pub. INC (1926). [ 1 ] : Stochastic integral. Proc. Imp. Acad. Tokyo, vol. 20, 519-524

[ 2 ] : Theory of Probability. (In Japanese). Tokyo, (1953) . C 3 3 : Stochastic process. Tata Institute Note, Bombay, (1959). S. It6 C I ] •' On Hellinger-Hahn's Theorem. (In Japanese). Sugaku, vol. 5 no. 2, 90-91 (1953). K. Karhunen £ 1 ] : Uber die Struktur stationarer zufalliger Funktionen. Arkiv for Matematik, 1. nr. 13, 144-160 (1950). P. Levy [ 1 ] : Processus stochastiques et mouvement Brownien. Gautier-Villars, Paris, (1948). [ 2 ] : Le mouvement brownien. Memorial des Science Math, f asc. 129, (1954) [_ 3 ] : Brownian motion depending on n parameters: The particular case « = 5 . Proceedings of Symposia in Applied Math. vol. 7, 1-20 (1957). [_ 4 ] : A special problem of Brownian motion, and a general theory of Gaussian random functions. Proceedings of the Third Berkeley Symposium on Math. Stat, and Prob. vol. II. 133-175 (1956). [ 5 ] : Sur une classe de courbes de I'espace de Hilbert et sur une equation integrate non lineaire. Annales Ecole Norm. Sup. torn. 73, 121-156 (1956). [_ 6 ] : Fonction aleatoires a correlation lineaire. Illinois Journal of Math. vol. 1, 217-258 (1957). [ 7 ] : Fonctions lineairement Markoviennes d'ordre n. Math. Japonicae, vol. 4, no. 3, 113-121 (1957). [ 8 ] : Sur quelques chasses de fonctions aleatoires. Journal de Math. pures et appliquees. torn. 38, 1-23 (1959). L. Schwartz [ 1 ] : Theorie des distribution I. Hermann & C'*, Paris, (1950). T. Seguchi and N. Ikeda [] 1 ] : Note on the statistical inferences of certain continuous stochastic processes. Memoires of the Faculty of Sci. Kyusyu Univ. Ser. A, vol. 8, No. 2, 187-199 (1954). T. Sirao [ 1 ] : On the continuity of Brownian motion with a multidimensional parameter. Nagoya Math. Journal, Vol. 16, 135-156 (1960). M. H. Stone [ 1 ] : Linear transformations in Hilbert space and its applications to analysis. Amer. Math. Soc. Colloq. Pub. vol. 15. (1932).

142

ANALYSIS ON HILBERT SPACE WITH REPRODUCING KERNEL ARISING FROM MULTIPLE WIENER INTEGRAL TAKEYUKI HIDA NAGOYA UNIVERSITY

and NOBUYUKIIKEDA OSAKA UNIVERSITY

1. Introduction The multiple Wiener integral with respect to an additive process with stationary independent increments plays a fundamental role in the study of the flow derived from that additive process and also in the study of nonlinear prediction theory. Many results on the multiple Wiener integral have been obtained by N. Wiener [14], [15], K. It6 [5], [6], and S. Kakutani [8] by various techniques. The main purpose of our paper is to give an approach to the study of the multiple Wiener integral using reproducing kernel Hilbert space theory. We will be interested in stationary processes whose sample functions are elements in E* which is the dual of some nuclear pre-Hilbert function space E. For such processes we introduce a definition of stationary process which is convenient for our discussions. This definition, given in detail by section 2, definition 2.1, is a triple P = (E*, n, {Tt}), where M is a probability measure on E* and {Tt} is a flow on the measure space (E*, p) derived from shift transformations which shift the arguments of the functions of E. In order to facilitate the discussion of the Hilbert space L 2 = L2(E*, /x), we shall introduce a transformation T defined by the following formula:

(1.1)

M ( £ ) = J ^ e^VOiO/iCdaO

for

v

e U,

where < •, •) denotes the bilinear form of x e E* and £ e E. This transformation T from L 2 to the space of functionals on E is analogous to the ordinary Fourier transform. By formula (1.1) and a requirement that T should be a unitary transformation, JF = T{LI(E*, H)) has to be a Hilbert space with reproducing kernel C(£ — 17), (£, r;) G E X E, where C is the characteristic functional of the measure p defined by

(1.2)

C(S) = f^-tWx), 117

Reprinted from Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, eds. Lucien Le Cam and Jerzy Neyman © 1967 University of California Press, pp. 117-143

ZeE.

143 118

F I F T H B E R K E L E Y SYMPOSIUM! HIDA AND IKEDA

The first task in section 2 will be to establish the explicit correspondence between Li and JF. Another concept which we shall introduce in section 2 is a group G(P) associated with a stationary process P. Consider the set of all linear transformations {g} on E satisfying the conditions that (i)

C(gO = C(f)

for every

f e E,

and

(ii) that g be a homeomorphism on E. Obviously the collection G(P) of all such g's forms a group with respect to the operation (01*72)£ = gi(gz£)- The collection G(P) includes not only shift transformations Sh, h real, defined by (1.4) 0Srf)(O = *(* - h), but also some other transformations depending on the form of the characteristic functional. An interesting subclass of stationary processes is the class of processes with independent values at every point (Gelfand and Vilenkin [4]). Sections 4-6 will be devoted to discussions of some typical such processes. Roughly speaking they are the stationary processes obtained by subtracting the mean functions from the derivatives with respect to time of additive processes with independent stationary increments. In these cases the independence at every point can be illustrated rather clearly in the space J by using a direct product decomposition of it in the sense of J. von Neumann [10]. Furthermore, because of the particular form of the characteristic functional, we can easily get an infinite direct sum decomposition of ff: (1.5)

1F= £

0fF„.

n=0

Each ff„ appearing in the last expression is invariant under every V0, g e G(P) defined by (1-6)

(7„/)(Q = /(
/eJF,

that is, Vg{5n) C ?» for every g <= G(P). With the aid of these two different kinds of decompositions, we shall investigate 5 and discuss certain applications. In section 5, we shall consider Gaussian white noise, although many of the results are already known. To us, it is the most fundamental example of a stationary process. Here the subspace 3n corresponds to the multiple Wiener integral introduced by K. Ito [5] and also to Wiener's homogeneous chaos of degree n. Kakutani [7] expressed the subspace of L2(E*, ju) corresponding to SF» in terms of the product of Hermite polynomials in L2. These results are important in the determination of the spectrum of the flow {Tt} of Gaussian white noise. It should be noted that the L2 space for this process enjoys properties similar to those of L2(»S", a„) over an n-dimensional sphere Sn with the uniform probability measure an. As is mentioned by H. Yosizawa (oral communication), the multiple Wiener integral plays the role of spherical harmonics in

144 ANALYSIS ON H I L B E R T SPACE

119

1/2 OS", (E*) be the c-algebra generated by all cylinder sets in E*. If C(f), £ G E, is a continuous positive definite functional with C(0) = 1, then there exists a unique probability measure n on the measurable space (E*, 03) such that (2.1)

C({) = j

&

exp [i(x, Z)Wdx),

£6 E

(cf. Gelfand and Vilenkin [4]). In what follows we shall deal only with the case in which E is a subset of Rr, where R is the field of real numbers and T is the additive group of real numbers or one of its subgroups. Every element of E then has a coordinate representation £ = (£(<)> t eT). For every h, we consider the point transformation Sh defined by (1.4). Whenever we are concerned with the Sh's, we always assume that E is invariant under all of them. For each St we define a transformation Tt on E* as follows: (2.2)

Tt:TtxeE*,

(eT

with

(Ttx, £> = (x, St£)

for every £.

Obviously {Tt: ( e T } forms a group satisfying (2.3) (2.4)

TtT, = T.Tt = Tt+S, T0 = I

s,teT,

(identity).

The group {Tt; t e T} can be considered as a transformation group acting on E*. Let 03 (T) be the topological Borel field of T. DEFINITION 2.1. The transformation group {Ttlt e T } is called a group of shift transformations if Ttx = f(x, t) is measurable with respect to 03 X A3(T). The

145 120

F I F T H B E R K E L E Y SYMPOSIUM: HIDA AND IKEDA

triple P = (E*, ji, {Tt}) is called a stationary process if /J. is invariant under shift transformations Th t £ T. DEFINITION 2.2. The functional C(£), £ e E, defined by (2.1) for a stationary process P = (E*, n, {Tt}), is called the characteristic functional of P. Having introduced these definitions, we begin our investigation of stationary processes. For convenience, we assume the following throughout the remainder of the paper. ASSUMPTION 2.1. There exists a system {kVjn-i, in e E, which forms a complete orthonormal system in H. For the measure space, (E*,
(2.5)

({
m ip{x)W)^dx),


LEMMA 2.1. The closed linear subspace of L2 spanned by { e ' , ^ e J J } coincides with the whole space L2. This can be proved by using the uniqueness theorem for Fourier inverse transform (for detailed proof, see Prohorov [13]). The next lemma is due to Aronszajn ([1], part I, section 2). LEMMA 2.2. For any stationary process P = (E*, n, {Tt}) there always exists a smallest Hilbert space $ = 3(E, C) of functionals on E with reproducing kernel C(£ — v), (£, if) e E X E, where C(£), J e £ , is the characteristic functional of P. Let us denote the inner product in ff by (•, •)• We now state some of the properties of $ obtained by Aronszajn: (i) for any fixed £ e E, C(- - £) e ff; (ii) (/(•), C(- - { ) ) = / ( { ) for any / e JF; (iii) SF is spanned by {C(- — £), £ e E). From these properties and lemma 2.1 we can prove the following theorem. THEOREM 2.1. The transformation T defined by

(2.6)

M(£) =

^ W e H W

is a unitary operator from L2 onto 3\ In fact, the relation (2.7)

r ( ± aje-^A

(•) = ± a,C{- - fc)

shows that r preserves norm since this relation can be extended to the entire space. Note that (2.8)

(rl)(-) = C(-)

and define (2.9)

Lf = { ^ e L 2 , ( ( ^ , l » = 0}, SF*= {/;/eSF, (/(•), C(-)) = 0}.

146 ANALYSIS ON HILBERT SPACE

121

Then r restricted to L\ is still a unitary operator from L% onto J*. On the other hand, if we define Ut, t e T, by (2.10)

TJt: (Ut
for

then {Ut, t G T} forms a group of unitary operators on L2 satisfying UtU. = UsUt = Ut+S, (2.11)

U0 = I

s,leT,

(identity),

Ut is strongly continuous. Moreover, we can see that {Ut; Ut = T Utr~l, t e T} is also a group of unitary operators acting on JF. For simplicity we also use the symbol Ut for Ut. Further, we write Ut even when Ut (or Ut) is restricted to L% (or ff*). In connection with G(P) we can consider a group G*(P) of linear transformations g* acting on E* as follows: (2.12)

G*(P) = {g*; g*x e E* for every x e E*, (g*x, £> = (x, g£) holds for every x e E* with g e G(P)}.

From the definition we can easily prove lemma 2.3. LEMMA

2.3.

The collection G*(P) is a group with respect to the operation

(2.13)

{g\g%)x = g\[glx).

Also, (2.14)

(0*)- 1 = ( r 1 ) * .

REMARK 2.1. Detailed discussions concerning G(P) and G*(P) will appear elsewhere. For the related topics on such groups we would like to refer to M. G. Krein [9]. By (1.3) (i), it can be proved that every g* G G*(P) is measure preserving; that is, n(d(g*x)) = n(dx). Hence, by the usual method, we can define a unitary operator Vg* acting on Lz(E*, n) by

(2.15)

(V,**)(aO =
<7*eG*(P),

^ e L 2 (#*, M ).

Similarly, we define (2.16)

(VJ)(&

= f(s,Q,

?eG(P),

/eJ.

Obviously, {Vg*; g* e G*(P)} and {Vg; g e G(P)} form groups of unitary operators on Lz(E*, fi) and SF, respectively. We now have the following relation between Vg* and Vg, (2.17)

( r ( * W ) ) ( 0 = V„-,(T?)(£),

which is proved by the equations (2.18)

fEte^M9*xMdx)

=

jEte^*-^HxMdg*-lx)

^ E ,

122

FIFTH BERKELEY SYMPOSIUM: HIDA AND IKED A

Another important concept relating to stationary processes is the purely nondeterministic property. Let T« be a set of the form (2.19)

T ( = {s;s G T , s < t}

and (B( be the smallest Borel field generated by all cylinder sets of the form (2.20) {x; {{x, £,), ••• ,(x, £„)) e B", fc G E, supp fc) C T „ K K n , S > e <&(R")}. The subspaces of L2, L2(t) and L%(t) are denned by Li(t) = {
'

t G T,

U(t) = W;

t e T.

Corresponding to Li(t) and Lt{t), we can define SF(0 = @ {C( • - £ ) ; £ e # , supp (?) C T,} **(0 = ( / ; / e f f ( / ) , (/(•), C(-)) = 0}, where ©{ } denotes the subspace spanned by elements written in the bracket. Then we can easily prove the following proposition. PROPOSITION 2.1.

For every t G T,

(2.23)

U(t) ^ $(t),

(isomorphic)

Lt(t) ^ (F*(<),

(isomorphic)

and (2.24)

under the transformation r restricted to L2(i) and L%(t), respectively. DEFINITION 2.5.

(2.25)

If

L S ( - » ) = n IS(«) = {0} «GT

ifeen P = (E*, n, {Tt}) is called purely nondeterministic. This definition was given by M. Nisio [12] for the case where E* is an ordinary function space. By definition and proposition 2.1, P is purely nondeterministic if and only if /IOWS,

(2.26)

3* (-oo) =

O 5*(s) = {0} «GT

holds. We are now in a position to develop certain basic concepts relative to stationary processes. We would like to emphasize the importance of a stationary process with independent values at every point. DEFINITION 2.6 (Gelfand and Vilenkin [4]). A stationary process P = (E*; n, {Tt}) will be called a process with independent values at every point if its characteristic functional C(£), £ G E, satisfies (2.27)

C(fc + fc) = C(&)C(&),

whenever

supp (fc) fl supp (fe) =

0.

If E is the space X of C°°-functions with compact supports introduced by L. Schwartz, this definition coincides with that of Gelfand and Vilenkin. If E

ANALYSIS ON H I L B E R T SPACE

123

is the space s of rapidly decreasing sequences, then we have a sequence of independent random variables with the same distribution. PROPOSITION 2.2. IfPisa stationary process with independent values at every point, then it is purely nondeterministic. PROOF. F o r / e 5*(t) there exists a sequence {/„} such that l.i.m.„_^„ /„ = / and (2.28)

/ „ ( • ) = Z ain)C(- - &">),

&} GE,

supp (&>) C T«.

Since P is a stationary process with independent values at every point, we have (2.29)

/„(f) = £ ai*5C(f - #">) = C(f) E 4 n ) C ( - ^ > ) = C(f)/,(0) 4-1

*-l

for any J with supp (£) C T?. However, /„(£) has to be zero since (2.30)

/ n (0) = (/„(•), C(- - 0)) = 0

for /„ e 5*.

Thus, /(f) = 0. If / G Dtet $*(t), then /(£) = 0 for every f with supp (f) C TJ for every i. Hence, we have /(•) = 0. Now we can proceed to the analysis of the L2(E*, p) space arising from a stationary process P with independent values at every point. First we discuss polynomials on E*. The function expressed in the form (2.31)

v(x)

= P{{x, fc), ••• ,{x, £„»,

? ! , • • - , £„ G # ,

a; e #*,

where P is a polynomial of n variables with complex coefficients, is called a polynomial on E*. If P is of degree p, we say that £)\pn(dx) < oo for | G E and every integer p > 1,

(')

/

(ii)

Jm (x, £)M(*0 = 0 /or ever// f e # .

With these assumptions we see that the set M of all polynomials on E* forms a linear manifold of L2. Consequently, r(M) = {T(^); p e M} is defined and T(M) C 3\ DEFINITION 2.7. 4 n operator Di; £ e ii', ?s defined by (2.32)

(ZVX-) = 1-i.m. - [ / ( • + «0 - /(•)] e—0

«

i/ i/ie Ziww'i exists, D$ is called a differential operator, and its domain is denoted by SD(-Df). We define 2D as Dies SD(-Dj)LEMMA 2.4. 7/ P satisfies assumption 2.2, we Ziaye i/ie following: (i) C(- — f) e 20/or ewer?/ £ e -E, and n * - i A A " — f) belongs to 2D /or an?/ n and £i, • • • , £ „ e i?; (ii) T ( M ) C 3D; (iii) for any £i, • • • , £ „ e I? and any choice of positive integers kh • • • , k„, we have

124

(2.33)

FIFTH BERKELEY SYMPOSIUM: HIDA AND IKEDA

r- 1 { ( n x D%) C{ •)} (x) = (») £ « n (x,

fe>*,

x e £*;

(iv) (i) -1 i){ is a self-adjoint operator, the domain of which includes {(?(• — £); fe£} Ur(I); (v) /or any f e r(M) and £1, £2 e # , we have (2.34) PROOF.

(2.35)

A^/(-)=^A1/(-)By assumption 2.2, l.i.m. - ( - (e"<*.«> - l)e*<*'i> - z"(x, £)<*<*">)- = 0. *-»o

L«

J

Using r, the above relation proves that (2.36)

l.i.m.-{C( • + * £ + „ ) + C(- + „ ) }

exists and is equal to r{i{x, f )e'}. In a similar way, we can prove the second assertion using assumption 2.2. (ii). By assumption 2.2, exp {»' £ * . i (,{J;, f,-)} is differentiable infinitely many times (in L2-norm) with respect to U, • • • , tn-i and tn, and we have (2.37)

(i) _ Z *'' \d

E

k<

/dtf • • • &&) exp [i ± ti{x, &>}!

= n (a;, &>*'. The right-hand side belongs to L2, and mapping by T, we have (2.33). For assertion (iv), if / e T(M), we have ( i V X , ) = (DJ(-),C((2.38)

- „) = ( l i m J [/(• + ef) - / ( • ) ] , C(- -

v))

= l i m - { / ( , + eC) - / ( , ) } , e->0 «

(/(•), Dfi(-

- ,)) = lim - (/(„ - ef) - /(u)) = - ( # « / ) « ,

which prove that (i)~lD$ is self-adjoint. For (v) consider (/(, + efi + «&) - /(r, + «&) - f(y + cfc) + /(r?))/e2. Arguments like the above prove that D^ and D& are commutative. N. Wiener [15] discussed the following decomposition of JF(L2) for the case of Gaussian white noise. Consider a system K of elements of J defined by KH = \jl (2.39)

DblC(-);h,

U-1

/To = {(?(•)},

and

. . . , * » = 1,2, • • • } , J ^ K = U #*

n > 1,

n=0

where {£?}"= i is the system appearing in assumption 2,1, Since the system K„

150 ANALYSIS ON H I L B E R T SPACE

125

forms a countable set, we can arrange all the elements in linear order. We shall denote them by {gr*n)(-)}- Let P be a stationary process satisfying assumption 2.2 and let ffo be the one-dimensional space spanned by C(-), that is, SFo = KQ. Put f? = g^ and define (2.40)

0i = © { # " ; * = 1,2, • • • } .

Obviously, SFo and SFi are mutually orthogonal. Suppose that {5j}"lJ are defined and mutually orthogonal, consider #»> = P(^"-1

(2-41)

ma

J=0

n Ygi \ Vol"'

k = 1, 2, • • •,

/

where P(.) denotes the projection on (•). Then SF„ is defined by (2.42)

JFn = ©{#">;* = 1,2, • • • } ,

and it is orthogonal to E^-To1 0 JFy. This procedure can be continued until there are no more elementsfid""1"1'not belonging to E?-o © ^V- Finally (2.43)

K ( = the closure of K in JF) = £ ©
DEFINITION 2.8. The direct sum decomposition (2.43) is called Wiener's direct sum decomposition. THEOREM 2.2. Let P be a stationary process satisfying assumptions 2.1 and 2.2. If (2.44)

£ gt\-)/n\,

#>(•) = (Z>t)"C(-),

n=0

converges for every £,
(2.46)

n=0 C/.CJF,) = JF».

By assumption, £ n=0

(tj-fo {)"/»! = ^

( £ gf(-)/n\\

(gf'(-) = C(-)),

V =0

/

converges and the sum is equal to exp {i(x, £)}. Hence, (2.47)

C(- - f) = E

^ ^

On the other hand, by the construction of the 5n's we can prove (2.48)

g?> e £ © SFy.

Therefore, C(- - f) e E"-o © 3v which proves (2.49)

JFC £ n=0

since {C(- — £)} spans the entire J.

©Sn(CS),

151 126

F I F T H B E R K E L E Y SYMPOSIUM: HIDA AND IKEDA

The second assertion is easily proved noting that (2.50)

UjtO) =

?$(•)

and (2.51)

®(U,Kn)

= ©(#»).

Let us return to the group G(P). If (2.52)

Vs(5n) C SF„

for every

g e G(P),

we call the decomposition J = Y,n = o © ?„ invariant with respect to G(P). This concept is important in connection with the Wiener's direct sum decomposition. We shall discuss this topic in the later sections (4-6).

3. Orthogonal polynomials and reproducing kernels From now on we shall deal with the decomposition of L2(E*, n) and 5(E, C) associated with a stationary process with independent values at every point. First we consider, in this section, the simple case where E is a finite dimensional space. We can find a relation between the space with reproducing kernel and Rodrigues' formula for classical orthogonal polynomials. Such considerations will aid us in considering the case where E is an infinite dimensional nuclear space and will be preparation for later discussions. Let v be a probability measure (distribution) on Rx and C be its FourierStieltjes transform (characteristic function); that is, (3.1)

C(X) = JRI e°"v{dx),

X e R\

Appealing to Aronszajn's results [1] stated in lemma 2.2, we obtain the smallest Hilbert space & = 3(Rl, C), the reproducing kernel of which is C*(X — /*), X, n e R1. By theorem 2.1, there exists an isomorphism f which maps Z 2 = Li(v; R1) = {/; JRI \f(x)\2v(dx)} onto 5 in the following way: (3-2)

(f/)(X) =

\me^f{x)v{dx).

We shall examine this isomorphism f in detail in the following examples. It is more interesting to discuss the analysis on § rather than on L2, since, for one thing, the development of functions belonging to L2 in terms of orthogonal polynomials turns out to be the power series expansion in §. 3.1. Gaussian distribution. Consider the case where (3.3)

v(dx)

= v(x;
dx,

then (3.4)

C(\, O = /

e**v(x; a2) dx = exp (~

X2Y

152 ANALYSIS ON H I L B E R T SPACE

127

Choose Hermite polynomials (3-5)

Hn(x; „«) = ^ f ^

^

£

,(*; O ,

n = 1, 2, • • •

(llodrigues' formula), which form a complete orthonormal system in L2- The isomorphism f maps Hn(x) to the n-th degree monomial of X times C. In fact, (3.6)

(iHn(- • 0 ) ( X ) =
The proof of the formula (3.6) is as follows: (3.7)

(fff„(., ff '))(X) (—1)V2" /' f rf" 1 = — ^ y — / {exp (iXz)} -Mx; cr2)"1 ^ v{x; a 2 )j- v (x;
= ^ i " /BI exy (*'Xa:) { « ! " ^ f f 2 ) } d x = ^i"X"C(X; f f 2 ). More generally, we have (3.8)

U ( E o anHn{-; a 2 ) ) } (X) = (tQ

3.2. Poisson distribution. (3.9)

anon*") C(X: cr2).

Let v(da;) be given by

v(dx) = v(x, c)8s£dx) = -

g

^ e"",

z G &,

where $ c = {—c, 1 — c, 2 — c, • • •}. We obtain orthogonal polynomials with respect to the measure v(x, c)8s£dx), which are called generalized Charlier polynomials, by the following generalized llodrigues' formula (cf. Bateman and others ([2], p. 222, and p. 227)): (3.10)

pH(x, c) = (-c)'(v(x,

c))~lA1v(x - n,c) = Un^-\c)n\,

x G &,

where A" denotes the n-th order difference operator acting on functions of x. The relation (3.11)

C(X:c) = ( ea'v(x;c)ds£dx)

= £

e^v{x, c)

= exp {eiXc — 1 — i\c} is easily obtained. Now put (3.12)

Qn(x,c) =

-\-J\(x,c);

then we get the orthogonality relation for Qn: (3.13)

Q„(x, c)Q c)v(x, c) = S„,m, £Z Q„(x, c)Q„«(a:, m(x, c>(z,

w, m = 1, 2, •

xGS,

Every Pn, of course, belongs to L2(v, II1), and it is transformed by f into

153 128

FIFTH BERKELEY SYMPOSIUM: HIDA AND IKEDA

(?P,(-, c))(X) » c"(eft - 1)-C(X: c).

(3.14)

This is proved as follows: (3.15)

(fP,(-, c))(X) = / f i i e^(-c)»((K*, c ) ) - ^ ; ^ ! - n, c)>(z, c)«s.(dz) = (-c)»(-l)" £ = cn E

(A3e**M:r,c)

Z (—l) n_ro

I eimXeixMx, c)

= c ( e a - 1)«C(X: c). Note that the last expression is of the monomial form of (eiX — 1) times C. 4. Stationary process with independent values at every point This section is devoted to the study of the general theory for a certain class of stationary processes with independent values at every point. Let P = (E*, ix, {Tt}) be a stationary process, where T is a set of real numbers and E is the function space S in the sense of L. Schwartz, and let its characteristic function be given by

C(Q = exp f (

4

'

1

}

«(£«)) dt, / • -

a(x)

= {-**x*)/2 + J _ (e*» - 1 - j ^ ) l-±^ #(«).

Here 0 < a2 < oo and dfi{u) is a bounded measure on ( — °°,°°) such that dp ({0}) = 0. Obviously, P satisfies (2.27); that is, it is a stationary process with independent values at every point. For the moment let us turn our attention from the flow {Ut, t real} to the direct sum decomposition of £F = 5(8, C) mentioned in section 1. Define K,{Z, n) = exp ( / «({(0) dt) exp ( / a ( - „ ( 0 ) dt) = C(£)C(-„), (4.2)

-Ki& v) = / «(£(0 - i?(0) * - / «(*(*)) dt - f a(-ij(0) dt, #,tt, i») = i (#i& u))p,

P > 2,

ir '

**(*, *) = r* #»(£, ij)(JCitt, *?))p, p!

p > 0,

it, e S.

Note that C(£ — r?) = E " - ofcj,(£,??). We then prove the following lemma using the fact that a(x — y) is conditionally positive definite (cf. Gelfand and Vilenkin [4], chapter 3). LEMMA 4.1.

The junctionals K0(£, »?), Kp{£, rj), and kp(£, rf), p = 0, 1, • • • ,

(£, rj) e S X S, are all positive definite and continuous.

154 ANALYSIS ON H I L B E R T SPACE

129

Again appealing to the Aronszajn's theorem (lemma 2.2), we obtain the Hilbert spaces JF, 5P and 5P, p = 0,1,2, •• • with reproducing kernels C, KP, and kp respectively. Consider subspaces Sp and 3> We use the symbol ®* to express the direct product of subspaces in the sense of Aronszajn [1]. Hereafter we use subscripts to distinguish the various norms. LEMMA 4.2. The space §p is the class of all restrictions of Junctionals belonging to $i ®* Si®* • • • ®* $i (p times) to the diagonal set $>p = {(£, • • • , f); £ e S}. The norm || • \\$r in Sp can be expressed in the form (4-3)

H/|k =

inf „. H/'lk ....®*. /=/'/« , wheref'/i>p denotes the restriction off to Sp. PROOF. By the definition of Kp and by Aronszajn ([1], section 8, theorem II) the assertion is easily proved. LEMMA 4.3. The space $p is the class of all restrictions of functionals belonging to §o ®* Sp to the diagonal set §2 = {(£, £); £ e S}. The norm \\ • \\$f can be expressed in the form (4-4)

||/lk =

inf

A

ll/'ll*®*,.

/-/'/Si

LEMMA

4.4.

77ie space SJ^, p = 1, 2, • • • , are mutually orthogonal subspaces

o/ff. PROOF.

Put

(4.5) J C , , ^ , v) = Kp{£, v) + K,& v), V * ?• Then the Hilbert space 5 p , g with reproducing kernel Kp,q(£, -q) is expressible in the form (4.6)

Sp,q = 5P © St.

To prove this assertion we first show that (4.7)

$Pn$g=

{0}.

Suppose p > q; then (4.8)

# p (£, „) = K9& v) ^ ^ f ^ " ' # , - , ( ? , u).

Consequently, ffj, is the class of all restrictions of functionals belonging to Sq ®* Sp-q to the diagonal set §2. Now suppose / e f , f | ^ and let {/*8)} be a complete orthonormal system in 9q. Since / e $>, it can be expressed in the form (4.9)

/(£) = £ gti&fP®,

ge9p-q

k= l

(remark attached to theorem II, Aronszajn ([1], p. 361)). But by assumption, / belongs to Sq. Consequently, all the gk's must be zero, which implies / = 0. Let us recall the discussion of Aronszajn ([1], part I, section 6). By (4.7), if fp e 9P, fq e 9t, then

130

F I F T H B E R K E L E Y SYMPOSIUM: HIDA AND IKEDA

Wfp + /
=

\\fp\\k + ll/«lk>

which imply Re (fp, /„) = 0. Similarly, we have Im (fp, fg) = 0. Thus we have proved (4.6) and the lemma. Further, by the proof of lemma 4.4, we can show the following. If 5>,„ is the reproducing kernel Hilbert space with kernel fc„(£, JJ) + kg(i-, y), then (4.11)

$p,q = J , © i F „

V ^ 1,

and if/ e ffp,a, then (4.12)

(/(•), M - , * ) ) = / , « )

is the projection of/ on J,,. LEMMA 4.5. The kernel Kp(£, 17) and /cP(£, 17), p > 0, are G(P)-invariant; that is, (4.13)

^2,(0?, gv) = Kp(£, 17),

&p(sr£, gy) = &„(£, 57)

/or every g e G(P). PROOF. It is sufficient to prove that K0 and Kx are G(P)-invariant. For A'0 this is easily proved by (4.2) and the definition of G(P). Concerning K\, we have (4.14)

KM,


-

— 00

f"

« ( ^ ) ( 0 ) dt

J — DC

/ " «(-(fi"7)(0) * = f"

a(flr({ - u)(0) * -

J — <M

/ — 00

- J a(-(gv)(t))dt

f

«(G/*)(0) ^

J — 00

= Xifer,),

since every a e G(P) keeps the integral / " „ a(1(0) dt invariant. Now we shall state one of our main results. THEOREM 4.1. The space J has the direct sum decomposition (4.15)

JF = E © 3 * , p= 0

and it is G(P)-invariant. The kernel kp(-, £) is a projection operator in the following sense: (4-16)

(/(•),*„(•,*)) = / , ( * )

is £Ae projection of f on 5^. PROOF. By lemma 4.4, (4-17)

± fcp({, „),

(?, , ) £ 8 X 8 ,

p-=0

will be the reproducing kernel of the subspace E"-o © 3> Noting that (4-18)

C«, ,) = £ *,(*, „), p= 0

and

156 ANALYSIS ON H I L B E R T SPACE

(4.19)

£ \p = n

131

kP(-,v)

we conclude (4.20)

J =

E©J» p= 0

(cf. Aronszajn [1], part I, section 9). The G(P)-invariantness of 'Sp comes from the definition of 5P and lemma 4.5. By (4.12) and the above discussions, we have the last assertion. Coming back to L2 space, we have the following decomposition: (4.21)

U = t

0 Uf

with

T(UP))

= fFp.

5. Gaussian white noise In the following three sections we shall discuss some typical stationary processes with independent values at every point. First we deal with Gaussian white noise, the characteristic functional of which is

C(£) = exp j - 1 J^ £(0 2
(5.1)

S e S>

namely the particular case where a(x) = — | z 2 in the formula (4.1). Consequently, Kp(^, -q) is of the form (5.2) Now put

Kp& v) = ± & v)> = ^j ( / _ „ f(0l(0 * ) ' • L 2 (^") = {F;F £ L2{Rv), F is symmetric}, . 1 F(
(5.3)

where TT denotes the permutation of integers 1, 2, • • • , p. Define 7J(£; F) by (5.4)

/•($; F) = £ _ • J $(*,) • • • i(tp)F(th

••• ,Q dh, ••• , dtp,

£ e S,

then we have (5.5)

I*v{Z;F) = I*v{$;F),

THEOREM

(i)

for every

g e S and

feL,(R»).

5.1. .For Gaussian white noise we have the following properties: Sp = {/(•);/(£) = /;(*; *0, f e L.Cfl")},

(ii)

(/*(•; F), / * ( • ; G))*, = p! / ^ / F«i, • • • , *p)G(*i, •••,*„) * • • -
PROOF.

(5.6)

Define L2(flp) and 5 P by

L2(R") = J F ; Ffo, ••• ,tp)=^-.t I

aMk) • • •&(«,),

p!fc-i

a* complex, & , • • • , ? , e S ^

157 132

F I F T H B E R K E L E Y SYMPOSIUM: HIDA AND IKEDA

and (5.7) Sp = {/(•);/(£) = ltd) F), F e U{R>)}. Then we can prove §p C $„. If F and G are elements of Ln(Rp) of the form

(5.8)

F= -

£ aMh) • • • h{tP),

G = -J-. E M*(«i) • • • Vk(tP),

p'*=i

p!fc = l

where c^'s and 6i's are complex numbers and £*, 17* e S, we have (5.9)

(/J(-;n«(-;G,))ff, » n _ 1 f " /* = Z Z a*by—, / "•' / &('i)---?t(^)i3(fi)-- - ^ ' ( y jfc=i,-=i PU -« J

dtvdtp

= p! / •_• • / F(ti, • • • , OGfo, • • - , « , ) <&• • -(ftp. Since Li{Rp) is dense in L2(RP), we can prove that JFp is also dense in 5> Indeed, (5.10)

$p = {/(•);/(*) = / S t t ; f ) , F e

U(R')}.

Thus, by (5.5), we get (i). The second assertion is easily verified using (5.5) and (5.9). Take a complete orthonormal system {iy}j°«i in L^R1) such that all the £j's belong to S (cf. assumption 2.1). COROLLARY 5.1 (M. G. Krein [9]). Define the functional (5.ii)

(512)

$&"•.•& (?) = 7 / ~ =

f J *&'•'•'••& J

:

ji, • • • , jg

n & O S

different positive integers, « positive integers such that T,kj=

kh • • • , kq

! p\

forms a complete orthonormal system in Sp. PROOF. The set of functional on S X • • • X S (p times) of the form

(5.13)

{ ^
is a complete orthonormal system in #i ®* • • • ®* #i (p times). On the other hand, we have

(5.14)

(*&;.\vft, *gf.v.^,)*, Vfcx!- • .ft,! Vk[l-•-k',1 Y.T. x

\PU

I ' ' ' / fjiCxd))- • •^.(
ir' J

—

M

y

X £ft(<,r(ibi + fe))' ' • ? A ( ^ ( P ) ) ' • , ^ ( 4 ' ( * 0 ) * • ' ^ ( ' ' ' ( P ) ) ^ 1 " '

' ^

158 ANALYSIS ON HILBERT SPACE

133

The right-hand side vanishes if ((ju fa), ••• , (jq, kt)) ^ ((/ 1; fa), ••• , (j'a, k'g)) (as sets) and is equal to (5.15)

1

fa\---kq\ &

-XJ6M'41-liXSm'dtT.-

i

otherwise, where ir(4) denotes the permutation of k integers. Moreover, if q ^ q', then (5.14) obviously vanishes. Thus we have proved the corollary. REMARK 5.1. The above result has already been proved by M. G. Krein ([9], section 4), although it is stated in a somewhat different form. COROLLARY 5.2. If Hn{x; 1) denotes the Hermite polynomial defined by (3.5) with c r = l , then

(5.16)

T-1{*Si,;;.v.-a(.)C(-)}(x) = Vv~Wfa\---kq\{i)-^

fl

Hkm{{x,il);i).

m= 1

PROOF.

(5.17)

The formula

f e'«> fi H^dx, &>; lMdx) JS*

m= l

EL [Ht.dx, &>; De^lm'JMdx) S*m=l

= fl m= 1

/

el(i£..JxM,)n(dx)

f JS*

e*<«U^Hkm{x;l)-l-e-Xidxc( V2TT

£

W(I.,-,I,)

(S, to) $) /

becomes

(5-18)

fc#V,

ntt,6JK7(C).

This proves (5.16). REMARK 5.2. From theorem 4.2 and the above result, we get the orthogonal development of the elements of L2 due to Cameron and Martin [3]. In the above discussion we use an important property of Gaussian white noise, that is, the equivalence of independence and orthogonality. For other cases discussed here, the multiple Wiener integral due to K. Ito [6] plays an important role. Let {Ij}"=i be a finite partition of T. Then we have n

(5.19)

C({) = n C(txi,),

€ e S,

3-1

where XJ, is the indicator function of Ij. Note that C(£x/,) has meaning even though fx/ m a y n ° t be in S. Now if we consider the restriction of C(£) to X(Ij); then (5.20)

C/#(f) = C({),

* e 0C(7y),

is a continuous positive definite functional. Therefore, we can follow exactly the same arguments as we did for C(£). Let us use the symbols ?(/,•), 9V{I,), and 5P(Ij) to denote the Hilbert spaces corresponding to JF, $p, and 5P defined for C(£). Then we have

159 134

FIFTH BERKELEY SYMPOSIUM: HIDA AND IKEDA

(5.21)

5 = n ®* SF(/y) y=i

by the formula (5.19). We can also prove (5.22)

fi

®*ff(/i) = n ® ?(/,•),

(isomorphic)

by J. von Neumann's theory [10]. Let <S>(Ij) be the smallest Borel field generated by sets of the form (5.23)

{x; (x, £) G B} £ e 3C(J), B is a one-dimensional Borel set,

and let Lz(Ij) be the Hilbert space defined by (5.24)

Li(Ij) = {
Then by (5.21), n

(5.25)

L2 = n ®* Li(Ij). y-i

Because of the particular form of C(£), we can prove that (5.26) l.i.m. (x, £,> g—• w

1

exists if £g tends to x/, hi L2(R ) as g —* a>. We denote the above limit by (x, x/,). We are now in a position to define the multiple Wiener integral of K. Ito. Let F(ti, • • • , ( , ) be a special elementary function (see K. ltd [5], p. 160) defined as follows: _ J a , v . . , „ , for {th • • • , tp) e Th X • • • X Tir, (5.27) F(th ...,tP)= l0) othemise where the TVs are mutually disjoint finite intervals. For such F, Ip(x;F) defined by (5.28)

IP(x;F)=

£

a,-,,...,,-, LT (x, Xr«>.

ti,- • -.t'p

y=i

is

This function satisfies the following properties (5.29)-(5.32): for any two special elementary functions F and G, (5.29) (5.30)

Ip(x; F + G) = Ip(x; F) + Ip(x; G), IP(x;F)

=

Ip(x;F),

where F is the symmetrization of F; (5.31)

/?(#; F) e L2 for any p and any special elementary function F,

and

«/*(*; f), h(x; G)» = P!(^, (?W), (o.oz) «/,(*; F),/,(a;;(?)» = 0, if p ^ g . The map Ip can be extended to a bounded linear operator from L2(RP) to L2, which will be denoted by the same symbol Ip. The integral Ip(x;F) is called the multiple Wiener integral. It is essentially the same as that of K. Ito except that we can consider complex L2(RP) functions as integrands.

160 ANALYSIS ON H I L B E R T SPACE THEOREM

5.2.

(5.33)

135

For every F e L2(RP), we have IP(x; F) = ( i ) ' r » { I ^ ; F)C(-)} (as).

PROOF. If F is a special elementary function, (5.33) is obvious by the definition of J* and T. In fact, if F is defined by (5.27),

(5.34)

/*(£; F)C®=

E

a*,...,,-, H <€, xr„>.

ii,- • • ,iP

y=i

Hence, we have (5.35)

=

T-I{H(-;F)C(-)}(X)

E . a*,...,,-, H [ r - ' { ( ' , X ^ ( . ) } ] . i i , - • • ,ip

Since

_1

T {((-,

(5.36)

J' = 1

-1

XT,)C(-)}(aO = ( i ) ^ , xr,)> the above formula is equal to (i)-p

Z

a,-,....,,-, n

ii,- - -,i p

(i,xr,>.

y=I

Such a relation can be extended to the case of general F. S. Kakutani [7] also gave a direct sum decomposition of L2 using the addition formula for Hermite polynomials. It is known that Kakutani's decomposition is the same as that obtained by using multiple Wiener integrals. Conversely, this addition formula can be illustrated by using the decomposition of ff. This was shown by N. K6no (private communication) in the following way. Let I, 11, and 72 be finite intervals such that I = h + h; then (5.37)

1 <•, Xi)"C(• ),= Z o 1 <•, xiWi•

xid ^ 4 ^ , <• > xi^~hC{• x/,)C( • xi').

Noting that (5.38)

^ ( - . X / W x i , ) 6 *(/,), C(-x/0

y=l,2,

GSF(/«)

and that (5.25) holds, we have (5.39)

r-i { i <•, x / W x / . ) Tjnhfc)] <•, x^-*C(- X /,)C(-x/.)} CO = ri; 1 {^<.,x/ 1 ) , : C(-x/ 1 )}(a:) • ''A1 { ( ^ T f c ) ! <•> x/«>-*C(-xi,)} W-^UCC-xyO} W ,

where 17 denotes the mapping from L2(7) to SF(J) which is similar to T. Here each factor of the right-hand side is expressed in the form (5.40)

rlS j ± (•, x/ I W(-x/ 1 )} (*) = (*)-*ff*«a-, X/.); |/i|) e

(5-41)

r** -j ^ 4 f c ) j <•> ^ ) - ^ ( • x/,)}- (^)

U{h),

= (i)-»+*Hn_k((x,xi*);\h\)

eU{h),

161 136

(5.42)

F I F T H B E R K E L E Y SYMPOSIUM: HIDA AND IKEDA

Tf.HCC-x/OK*) = 1

eU{I°),

where |/| denotes the length of the interval / . On the other hand, since (5.43)

r-i ( i <-, »>-C(-)) (x) = Hn{{x, Xr); \I\),

we get (5.44)

Hn((x, xi); \I\) = t

Hk((x,

x/l);

\h\)Hn-k({x,

x/,); |/.|).

4= 0

Therefore, (5.45)

Hn(x + y; \h\ + \h\) = £) #*(*; [/i|)ff»_*(i/; |/,|) * =o

for almost all (x, y) e R2 with respect to the Gaussian measure,

(5 46)

-

-^^rA-m-m\dxdy-

Since Hn(x; a2) is a continuous function of x, (5.45) is true for all (x, y) e .R2. Indeed, (5.45) is the addition formula obtained by S. Kakutani [7]. Let us further note that N. Kono has shown that (5.45) can also be proved by using the Gauss transform defined by (5.47)

Hy) = /g,

2/ 6 S*.

This transformation is well-defined for polynomials. Since the transformation is bounded and linear, and since polynomials form a dense set in L2(S*, fx), we can extend (5.47) to all of L2(S*, fi). Let lp be the Gauss inverse transform of
(
for

ft^ei,(S*,ri

and

jeS*.

By simple computations we can prove the following: if p(a;) = H„((x, £i), 1) and

(5 49)

-

W ) (

fcn,mHn+m({y,

£), 1)

^ K ^ ¥ ( < / )

More generally, we can prove that if

(5.50)

L(2m),

for

& = £2 = £,

for

<6,6) = 0.

then

TK)

This operation becomes simpler when it is considered in SF. We shall use the same symbol ° to express the corresponding operation, namely, (5-51)

fog

= r ( ( r - y ) . (r"V))

for

/ , g G JF.

Recalling that the SF„ appearing in Wiener's direct sum decomposition of JF is r(L2n)), we have the following proposition. PROPOSITION 6.1. The spaces {^fn}n-o form a graded ring with respect to the operation •>. (For definition, see Zariski and Samuel [17], p. 150).

162 ANALYSIS ON H I L B E R T SPACE

137

6. Poisson white noise In this section we shall deal with Poisson white noise, which is another typical stationary process with independent values at every point. Our goal is to find the explicit expressions for E?, Sp, and Jp and also to look for relations between the multiple Wiener integrals and the Charier polynomials. Since Poisson white noise enjoys many properties similar to those of Gaussian white noise, we shall sometimes skip the detailed proofs except when there is an interesting difference from Gaussian case. The characteristic functional of Poisson white noise P is given by (6.1)

C(0 = exp U^

(c*« - 1 - tf(0) <&}»

S e S,

that is, a(x) = (eix — 1 — ix) in the expression (4.1). Hence, Kp(%, rj) is expressed in the form

(6.2)

KV{1, u) = ^ (J_ ^ PWWMt))

dtj,

f, T, e S,

where P{x) = eix — 1. For F e L2(RP) we define J%(£; F) by (6.3)

J*(f; F) = f-jm j Pim)

• • -PWP))F{h,

••• ,tv)dtv-

-dtp.

Obviously, (6.4)

JM;F)

= JP(Z;P),

£eS,

still holds (cf. (5.30)). THEOREM 6.1. For Poisson white noise, we have (i) (ii)

$ , = {/(•);/(£) = Jifa F), F e Lt(R*)} (Jt(-;F),JU-;G))*,

= p! fjaf

F(th ••• , QG{h, ••• ,tp) dh- --dtp

for any F,G G L^R"). PROOF. The proof is nearly the same as that of theorem 5.1. Thus, we shall just point out the necessary changes. The spaces Lt(Rp) and Sp have to be defined in the following way:

U{R') = U; F(h, • • • , Q =p!fc —.=t i o*P(&(«0)- • -PiUtp)); I (6-5)

aic

S, = {/(•);/© =

complex, & e S

Jt(Z;F),feURv)}.

p

If we prove that L2(fl ) is dense in L2(RP), then the rest of the proof is exactly the same as that of theorem 5.1. To do this, note that the totality of all linear combinations of functions such as XTI(*I)- • •XTp(,tJ>) with disjoint finite intervals {T,},f-i is dense in L2{RV), and also note that the fact that (6.6)

[•"•• [ E akP(Uh))J -"> J

k

• •P(fc(*i))xn(*i)- • -XTr(fp) dh- ••dtp = 0

138

F I F T H B E R K E L E Y SYMPOSIUM: HIDA AND IKEDA

for any choice of {a*} and £*'s in S implies that (6.7) xn(fd---XT,(tp) = 0, a.e. v We can therefore prove that L^R") is dense in Li{R ). The direct product decomposition of 5 and L2 is the same as in section 5. For any finite partition {Ij}J=i of T, (6.8)

C(f) = H C(&/,),

* e 8,

still holds. Therefore we have, using the same notation, (6.9)

JF = H ®* 5(1 i),

(6.10)

L2 = n

®*L 2 (/,).

Moreover, we can define the multiple Wiener integral with respect to Poisson white noise similarly. First note that (x, xi) is defined as an element of L2. If F is a special elementary function given by (5.27), then Jp(x; F) is defined by (6.H)

JP(x;F)=

£

a.v-A. H

(x,XTit).

The map J¥ can be extended to a bounded linear operator from Li(Rv) to L2 as was done in section 5 (cf. K. Ito [6], section 3). THEOREM 6.2. For every F e Li{R"),

(6.12)

JP(x;F) = T-H/JC-J^CCOXX).

PROOF. This proof is also the same as that of theorem 5.2, except for the following relation:

(6.13)

r((x,XTd---(x,XT^KQ

= n j=l

f

PWMdtjCti).

J I'i

From the last theorem we can show that

(6.14)

JF = £ 0 JF,

is nothing but Wiener's direct sum decomposition. This fact can also be proved using a certain addition formula for a one-parameter family of generalized Charier polynomials: let v{x, c) be given by cx+c

(6.15)

v{x, c) =

r

c

^~~ e~%

x = - c , 1 - c, 2 - c, • • • ,

and let (6 16) Pn(x, c) = n!PJ(a:, c) = {-cY(v{x, c))~lAy{x - n, c), where A" is a difference operator of order n; then the formula is (6.17)

P'\x + y, c) = £ Pt(x, cOPJ-tte, e2),

ci, c2 > 0, c = d + c2.

ANALYSIS ON H I L B E R T SPACE

139

7. Concluding remarks The theorems given in sections 5 and 6 extend to generalized white noise. Furthermore, we shall show that a sequence of independent identically distributed random variables can be dealt with in our scheme. We do not take up detailed discussions but summarize some of their properties. 7.1. Generalized white noise. We now discuss the stationary process with the characteristic functional C(0 = exp<

aW))d*h

l J

(7.D

-

*es> J

«W = i_.( eta *- 1 -TTb) 1 -^^" ) ' which is the one obtained from (4.1) by eliminating the Gaussian part — (a2/2)xi. Then Kp(l-, rj) is expressible as follows: (7.2)

Kp(i, „) = i Uj2

Pmu)P(v(t)u) du(t, «) j

where di/(t,u) = dt1+% d(3(u). We introduce the following notations: Dp = R2p, (7.3)

dvp = dvX ••• Xdu (p times),

L2(DP; mp) = i F; F is square summable with respect to —f dmp Li(Dp; mv) = {F; F e L2(DP; vv), F({th u{), ••• , (tp, up)) = F((trm,

«r(i)), • • • , (tT(ph ur(p))) for any permutation *•}.

Define M%{$; F) by (7.4)

M*(£; F) = j"af

P(f (ti)«i) • • -Pm>vW((h,

«i), • • • , (tp, uv))

X di>(^> Wi)- • -dv(tp, up) using the same technique as in sections 5 and 6. Then we have (7.5)

M*(£; F) = M*(£; f),

Ze s-

where (7.6)

H(h, ud, ••• , (tP, uP)) = —, E F((Um, u*m), ••• , (<*«> «*«))•

For generalized white noise with characteristic functional (7.1), we have the following results: fr„ = {/(•);/(!) = M*P(V, F), F G U{DP, up)} (i) Mm;F) = M^;F); (ii)

(M*(-; F), M*(-; G))$p = p ! | | " _ / ( ( f c , «i), • • • , (tp, «„)) X G((«i, Mi), • • • , («„, M P )) rfl/p ((f 1; M , \ • • • , (<„, « p ) ) .

165 140

F I F T H B E R K E L E Y SYMPOSIUM: HIDA AND IKEDA

Let us emphasize some of the important differences from Gaussian or Poisson white noise. First we cannot expect that the decomposition JF = 22P=o © $P, where SFj corresponds to the Sp appearing in (i), will be the Wiener's direct sum decomposition. However, r - 1 ^ ) coincides with the multiple Wiener integral introduced by K. It6 [6]. The next remarkable thing concerns the direct product decomposition. Let {Jy}jLi and {Jk}f=i be finite partitions of T and R1, respectively, and define (7.7)

C(f; /,- X Jk) = exp { £ f^ (e*<0- - 1 - ^

^

Recall C(£; I, X Jk) defines the subspaces SF(Jy X Jk), Sp(Ij X / * ) . Since n

(7.8)

l

-±^

d${u) dt}«

3>(/y X Jk)

and

m

C(£) = n n C(£;IyX J»), 3=1*=1

we have (7.9)

? = n n ®* ff(/y x /*). 3 = 1 Jb-l

Now we note a connection with K. Ito's multiple Wiener integral. It seems to be difficult to start in the same way as in sections 5 and 6 by introducing (x, xi) in L2. However, if we consider SF, we can proceed by defining for finite intervals / and J, (7.10) (7.11)

Af *({; IXJ) M(x;I

XJ)

= {J,

Pm)u)u

= T-\M*{-;I

dv(t, u), X

J)C{-)).

M(x; •) can be considered as a random measure as in K. I to ([6], section 3) and using it, we can define the multiple Wiener integral IP(F). Let us denote it by Mp(x; F). Then, for every F e L 2 (D P , mp) we can easily prove that (7.12)

r~KM*p(-; F)C(-)) = Mp(x; F).

Rather than discuss the group G(P) in detail, we shall just give a simple example. EXAMPLE. Consider the case where (7.13)

a(x) = \x\>,

0 < 6 < 2.

This corresponds to the symmetric stable distribution with exponent 0. A transformation g on E belongs to G(P), that is, (7.14) if and only if

(7-15)

C(rt) = Ctt),

fe E

/_"j(00(ON* = /_"j*(OI'd*.

Then g, (7.16) (0{)(0 = ct(cH), is an example satisfying (7.15).

c > 0,

ANALYSIS ON H I L B E R T SPACE

141

7.2. A sequence of independent random variables. Consider a stationary process P = (E*, /x> {Tt}) with independent values at every point, where E = s = (£ = {£*} * - - « ; f* real) is the space of rapidly decreasing sequences and T is the additive group of integers. A system of independent identically distributed random variables arises in the following way. Take a sequence {£„}"_ _ . of sequences, (7.17)

fc,

= © ? . - e s , f , = 8n,k.

Since the £„'s have disjoint supports, the (x, f„) = Xn(x), mutually independent. Further, we have (7.18)

UtXn{x) =

— oo < n < °°, are

= X, + ( (z).

Z»(2VE)

In view of the above, T = T\ is called a Bernoulli automorphism and P is called a stationary process with a Bernoulli automorphism T. If C(z) is the characteristic function of Xn(x), that is, (7.19)

C{z) = / s , * • * . « „ ( & ) ,

then the characteristic functional C of P is expressible in the form (7.20)

C({) =

n

Cm,

$={{*}»-__. e « .

&= — «

We can now form the Hilbert space EF = JF(s, C) with reproducing kernel C given by (7.20). The direct product decomposition and the direct sum decomposition of CF can be done in section 4. We would like to mention two particular cases of stationary processes with Bernoulli automorphisms. (a) The Gaussian case. Let P = (s*, /x, {Tt}) be the stationary process with a Bernoulli automorphism. Suppose that the characteristic functional of P is given by C(0 = exp {-m\\2},

(7.21) 2

fe«

2

where ||f|| = E"=-°° (£*) . In this case, the subspace EFj, of SF(s, C) turns out to be the following: (7.22)

SF, = {/(•); /(i») =

oO'i. • • • - i r V 1 • • • V'Ciri),

E

v = M " - - » e s, oO'i, ••• ,j,)e URp)\> where &CSP) is defined by (7.23)

4 ( # p ) = {oO'i, • • - , ; , ) ; V.

|o0'i, • • • , Jv) 12 < ° ° , «0'i> • • • , J P )

£ h, •••tit—

—"

is symmetric with respect to jk's >•• If the fk(-), k = 1, 2, in JF are given by

167 142

F I F T H B E R K E L E Y SYMPOSIUM: HIDA AND IKEDA

(7.24)

fk(v) =

ak(jh • • • ,j,)r,*- • -vj'C(v),

t ;'i.

• • • • 3 p =

-

k=l,2,

*

then Ave have the following: (7.25)

(fafch

= p!

E

aiO'i, • • • ,Jp)
Actually J = E"=o © fFP is the Wiener's direct sum decomposition. Thus, the subspace Lip) of L2(s*, /*)> corresponding to SFj,, is expressed as (7.26)

LT = © < n Hpt(Xqk(x); 1); {g*} different, Z P* = P U=i

i

(b) Poisson case. Consider the stationary process P = (s*, y., {Tt}) characteristic functional is (7.27)

C(f) = exp |

f; _ (e*" - 1 - if')}'

whose

€ = {€'}"-— e s-

Of course, P is a stationary process with a Bernoulli automorphism. The interesting thing is that L'2P) is spanned by elements of the form (7.28)

E

a,,,,...,„. fl Q w C M z ) ; l ) , E P* = P,

PI, • • ' , P n

&= 1

ft=

1

where Q„ is the function defined by (3.12). Note that the Qn's form a complete orthonormal system in L2(Si, dv(x, 1)) (for notation, see (3.9)). We can also prove that (7.29)

ffp

= {/(•);/« =

E

a{ji, • • • ,J,) fi (e*-'" - 1)C(,), a(jh ••• ,jr)e

URp)

Although the expression of/(•) in (7.29) is quite different from that in (7.22), we still have the same formula for the inner product, that is, (7.25). REFERENCES [1] N . ARONSZAJN, "Theory of reproducing kernels," Trans. Amer. Math. Soc, Vol. OS (1950), pp. 337-404. [2] H. BATEMAN et ah, Higher Transcendental Functions, Vol. 2, New York, McGraw-Hill, 1953. [3] R. H. CAMERON and W. T. MARTIN, "The orthogonal development of non-linear funct i o n a l in a series of Fourier-Hermite functionals," Ann. of Math., Vol. 48 (1947), p p . 385-392. [4] I. M. GBLFAND and N. Y A . VILENKIN, Generalized Functions, IV, New York, Academic Press, 1964. [5] K. I T 6 , "Multiple Wiener integral," J. Math. Soc. Japan, Vol. 3 (1951), pp. 157-169. [6] , "Spectral type of the shift transformation of differential process with stationary increments," Trans. Amer. Math. Soc, Vol. 81 (1950), pp. 253-263. [7] S. KAKXJTANI, "Determination of the flow of Brownian motion," Proc. Nat. Acad. Sci., Vol. 36 (1950), pp. 319-323.

ANALYSIS ON H I L B E R T SPACE [8]

[9]

[10] [11] [12] [13]

[14] [15] [16] [17]

143

, "Spectral analysis of stationary Gaussian processes," Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley and Los Angeles, University of California Press, 1961, Vol. 2, pp. 239-247. M. G. K R E I N , "Hermitian-positive kernels on homogeneous spaces," I and I I , Amer. Math. Soc. Transl, Ser. 2, Vol. 34 (1963), pp. 69-164. (Ukrain. Mat. Z., Vol. 1 (1949), pp. 64-98; Vol. 2 (1950), pp. 10-59.) J. VON NEUMANN, "On infinite direct product," Compositio Math., Vol. 6 (1939), pp. 1-77. , "On rings of operators," Ann. of Math., Vol. 50 (1949), pp. 401-485. M. NISIO, "Remarks on the canonical representation of strictly stationary processes," J. Math. Kyoto Univ., Vol. 1 (1961), pp. 129-146. Yu. V. PKOHOBOV, "The method of characteristic fuiictioiials," Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley and Los Angeles, University of California Press, 1961, Vol. 2, pp. 403-419. N . W I E N E R , " T h e homogeneous chaos," Amer. J. Math., Vol. 60 (1938), pp. 897-936. , Nonlinear Problems in Random Theory, Technology press of M.I.T., Wiley, 1958. N . W I E N E R and A. WINTNER, "The discrete chaos," Amer. J. Math., Vol. 65 (1943), pp. 279-298. O. ZARISKI and P . SAMUEL, Commutative Algebra, II, Princeton, Van Nostrand, 1960.

Reprinted from JOURNAL OF MULTIVARIATE ANALYSIS All Rights Reserved by Academic Press, New York and London

Vol. 5, No. 4, December 1975

The Square of a Gaussian Markov Process and Nonlinear Prediction* T . H l D A AND G . KALLIANPUR Nagoya

University

and University

Communicated

of

Minnesota

by the Editors

A n explicit f o r m u l a is o b t a i n e d f o r t h e n o n l i n e a r p r e d i c t o r of Y(t) = X(t)2 — E(X{ty), w h e r e X(t) is a n iV-ple G a u s s i a n M a r k o v p r o c e s s .

1.

INTRODUCTION

Let us begin with some general remarks about nonlinear prediction. Let X(t), t e [0, oo) be a stochastic process, EX{t) = 0 and EX(t)2 < oo for each t and let Bt(X) = a{X(u), u ^ i). Suppose it is required to predict the value of the process Y(t) = 6[X(t)] - E[6(X(t))],

(1.1)

given US(X), s < t, when 6(x) is a real-valued Borel function of x. Clearly, the best, i.e., least-squares (nonlinear) predictor Y^t, s) of Y(t) among all US(X)measurable, square integrable random variables is given by f1(t,s)=E[Y(t)\Bs(X)].

(1.2)

If the values of {X(u)} are not observed but instead, one observes only the Y-process, the best predictor based on B S (F) is Y,{t, s) = E[Y(t) | US(Y)].

(1.3)

R e c e i v e d M a r c h 3 1 , 1975. A M S 1970 s u b j e c t classification: P r i m a r y . Key words a n d phrases: Nonlinear prediction; Gaussian Markov process; Multiple W i e n e r integral. * W o r k supported in part b y National Science Foundation G r a n t G P 30694X.

451 Copyright © 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.

452

HIDA AND KALLIANPUR

Consider now the nonlinear prediction problem for Y(t) (given by (1.1)) when X(t) is a Gaussian, iV-ple Markov process which has a canonical representation of the form X(t) =

f F(t,u)dB(u),

(1.4)

where B(t), t e [0, oo), is the standard Brownian motion and where the following assumptions are made. X{t) is (N — 1) times differentiable in quadratic mean and there exists an iVth order differential operator Lt such that LtX(t)

= (d/dt) B(t)

(1.5)

in the sense of distributions. Furthermore, the kernel F(t, u) in (1.4) is the Green's function of the operator Lt and is given by N

F(t, u) =

£/<(*)&(")

if

u
i-l

0

if

(>«,

where t h e / / s and the g / s form a fundamental system of solutions oiLtf = 0 and Lt*g = 0, respectively, Lt* being the adjoint of Lt. For further details concerning these assumptions we refer the reader to [1, Theorem II.6, p. 138]. Writing mAt) for the Wiener integral §o£i(u) dB(u), we have

*(0=Z/<(')«*(*)•

(1-7)

2=1

Now it can be shown that for every i = 1,..., N, mAt) is a linear combination of the random variables X{i)(t),j = 0,..., N — 1, Xli,(t) being the/th-derivative of X(t) in quadratic mean. It follows from (1.7) that if 6{x) = x, i.e., Y(t) = X(t), the best predictor, Y^t, s) is a linear function of the N random variables Xu)(s) (j.= 0,..., N — 1). T h e situation is considerably more complicated when Y(t) is a nonlinear function of X(t). The predictor Y-A^t, s) (defined by (1.2)) is then measurable with respect to the a-field a[X(u), X'(u),..., Xw~1){u), u < s]. Furthermore, if only the values of the F-process are observed, the question arises whether the optimal predictor in this case, viz, Y2(t, s) can be expressed (in analogy with the linear problem) in terms of Y(s), Y'(s),..., F (JV-1) (s). It should be noted that from the properties of X(t) and the kernel F(t, u) assumed above, it easily follows that Y(t) is also differentiable (N — 1) times in quadratic mean, whenever d is smooth.

453

NONLINEAR PREDICTION

It is the aim of this paper to solve the above-mentioned problem for the square of the Gaussian process, i.e., where Y{t) = X{tf - E(X{tf),

(1.8)

and X(t) satisfies the assumptions made in (1.4)—(1.6). It will be shown, in fact, that in this case Yx(t, s) is measurable with respect to B S (F) so that it coincides with the predictor Y2{t, s) and that if s ^ t,

?i(*, *) =

I

m m ) * « ™ , Y'(s),..., 7 <"-»(,); s),

(1.9)

where the 3>j/s are rational functions. Corollary 1 and Theorem 1 are applied to two illustrative examples at the end of the paper.

2. T H E SPACE £%(t) OF NONLINEAR FUNCTIONS OF THE F-PROCESS

It is easy to see from the definition of X(t) given in (1.4) that Y(t) can be expressed as a double Wiener integral in the sense of Ito [3, p. 163]: Y(t) = f f F(t, u)F(t, v) dB{u) dB{v) •'o •'o

=

I

MM)

i.j=l

f fgi(u)gi(v)

J

0

dB(u) dB(v).

(2.1)

J

0

Writing Mi&) = f f

gi{u) gj{v)

dB{u) dB(v),

we see from the definition of multiple Wiener integrals that Mtj{t)

= MH{t)

= U* f' [&(«) gfr) + &(«) gM dB{u) dB{v).

(2.2)

Obviously, Mtj(t) is a martingale relative to the increasing family of cr-fields Et = a{B(s); s < t}. We further note that, for every t, M y (*) is an element of the Hilbert space 3^2 °f double Wiener integrals with respect to B. Thus we are given \N{N + 1 ) martingales living in 3^ .

454

HIDA AND KALLIANPUR

Let &r(t) be the vector space of rational functions of Ym(t), 0 < k < N — 1, which are in J^2> a n d l e t Qxip) D e t n e linear vector space of all quadratic functions of Xu)(t), 0 < 7 < iV — 1, minus their mean values. Since Xw(t)'s are all Gaussian random variables, we see that Qx(t)C^.

(2.3)

PROPOSITION 1. For each fixed t, we have Qx(t)C<%r{t). Proof.

(2.4)

Each member of Qx(t) is a linear combination of variables of the form *<*>(*)*«>(*)-y*.i(0.

where yjc,i(t) = (dkJrljdik dsl) r(t, s)\t=s, r(t, s) being the covariance function of X(t). T o prove that such variables belong to &Y{i), we use induction with respect to n = k + I. T h e argument is as follows. Clearly, (i)

X{tf

- yjf)

= Y(t) belongs to 9tY{t).

(ii)

Next assume that X
for every

&,/<«•

(2.5)

Then, (a)

X^{t)X«\t)-7n+ia(t) = {dldt)(X^(t)

XV(t)

- YnM

- l*M(t)

X<w\t)

-

yn,l+1(t)]-

Since the class 3$Y{i) is closed under the operation djdt, we see that the above expression is in 8%Y{t) provided I < n, that is, X<«+»(t)X«\t)-yn+ia(t)e®Y(t). (b)

X^\t)

X«\t)

- y„ +1 ,„(i) = i(dldt)[X^(tf

-

y B>n (f)].

By assumption (2.5) and by the same reasoning as in (a) we can immediately see that the above quantity belongs to 0tY(i). (c)

By the use of the formula in (b) we have

jr<«+i>(f). -

=

yn+1_n+1(t) {n

[\\^[X W-YnAt)]+yn+Ut)f/xM(tY]~r«+i.n+i{t)-

455

NONLINEAR PREDICTION

Both the denominator and numerator belong to 3#y(t), as does the fraction. From (a), (b), and (c) it follows that (2.5) holds for every k, I < n + 1, which was to be proved.

3. T H E MARTINGALES

M{j(t)

The martingales Mtl{t) introduced in Section 2 play an important role in our approach. First we prove PROPOSITION 2. For each fixed t, M i 3 (i)'s, i ^ j , ore linearly independent vectors in 3^ . Proof. T h e integral representation theory of the members in #f2 (see, for example, [2]) tells us that there is an isomorphism between ^ and L?(R2) the space of all symmetric L2(7?2)-functions. We can therefore associate the L2(R2)function {&(«)&•(») + g,(u)gi(v)} X xioMu> v) w i t h 2M«(0- By the use of this representation we prove the linear independence. Suppose that £ a^Mifit) = 0, atj constant. (3.1) a This can be expressed as

I ««{&(«) g&) +ft(«)&(«» = 0

on L\{0, tf).

For any continuous function 99 supported by [0, t] we have Z atMsP, gi) gi(v) + (
on

[0, t],

where (95, g) — $
Z au(9>gi) + Z ai*l