2 DAVID DOUGLAS ENGEL

Motion is a function X(t,o)):]R x ft - 1R satisfying the following three

conditions:

(A) For any s t e H , (X(t,o) - X(s,o))) is a Gauss function

of weft with mean zero and variance m([s,t)).

(B) If t0t1t2t3o..tk then {(X(tjW) - X(t.^.u,)) } . ^ ^ _ _fc

is an independent system.

(C) X(0,o) = 0 for all 0 0 e ft.

When there is no chance for error the dependence on u will be suppressed

and we write X(t) = X(t,a)). The concept of Brownian Motion can be gener-

alized by setting X(I) = X(t) - X(s) when I is the interval [s,t). In

this form Brownian Motion is a function X(I,OJ): o f x ft - * ] R satisfying the

following three conditions:

(i) For any I e *} X(I,u)) is a Gauss function of a) with mean

zero and variancem(I).

(ii) If I ,...,1 are disjoint elements of 3 then {X(I.): l£j£k}

I K .

J

is an independent system.

(iii) If I = IfJI2 (disj) then X(I,u) = X(I ,0))+ X(I2,u) for a.a.u).

Remark: If I = U I. (disj) then (ii) and (iii) imply that X(I.u))=

i=l 1

n

2

L.I.M. 23 X(I.,u) where L.I.M. denotes limit in L -mean.

n •* °° i=l

This means that Brownian Motion is a (norm) countably additive

2

set function whose values lie in L (ft,®,Pr).

The Homogeneous Chaos

We now consider the polynomial chaos and homogeneous chaos introduced

by N. Wiener [9]. Let us denote by I P the set of all functions g(w) of

the form

g(u) = P(X(I1) X(I))

where P(u-,...,u ) is a polynomial of degree n and I ,...,I are elements

of y . This means that there exists real numbers a such that