64.6. WIENER PROCESSES, ANOTHER APPROACH 2215
which shows that the inverse Fourier transform of ν is 0. Thus ν = 0. To see this, let ψ ∈S,the Schwartz class. Then neglecting troublesome constants in the Fourier transform,
0 =∫Rp
ψ (t)∫Rp
eit·ydν (y)dt =∫Rp
∫Rp
ψ (t)eit·ydtdν (y) = ν(F−1
ψ)
Now F−1 maps S onto S and so this reduces to∫Rp
ψdν = 0
for all ψ ∈S. By density of S in C0 (Rp) , it follows that the above holds for all ψ ∈C0 (Rp)and so ν = 0.
It follows that for every B Borel and for every such description of W(h).
0 =∫
Ω
XXB (W(h))dP =∫
Ω
XXW(h)−1(B)dP
Let K be sets of the form W(h)−1 (B) where B is of the form B1×·· ·×Bp,Bi open, thisfor some p. Then this is clearly a π system because the intersection of any two of them isanother one and
/0,Ω = W(h)−1 (Rp)
are both in K . Also σ (K ) = F . Let G be those sets F of F such that
0 =∫
Ω
XXF dP (64.6.35)
This is true for F ∈K . Now it is clear that G is closed with respect to complements andcountable disjoint unions. It is closed with respect to complements because∫
Ω
XXFC dP =∫
Ω
X (1−XF)dP =∫
Ω
XdP−∫
Ω
XXF dP = 0
By Dynkin’s lemma, G = F and so 64.6.35 holds for all F ∈F which requires X = 0.
64.6.2 The Wiener ProcessesRecall the definition of the Wiener process.
Definition 64.6.6 Let W (t) be a stochastic process which has the properties that whenevert1 < t2 < · · ·< tm, the increments {W (ti)−W (ti−1)} are independent and whenever s< t, itfollows W (t)−W (s) is normally distributed with variance t−s and mean 0. Also t→W (t)is Holder continuous with every exponent γ < 1/2, W (0) = 0. This is called a Wienerprocess.
Now in the definition of W above, you begin with a Hilbert space H. There exists aprobability space
(Ω,F̂ ,P
)and a linear mapping W such that
E (W ( f )W (g)) = ( f ,g)