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