64.3. WIENER PROCESSES IN SEPARABLE BANACH SPACE 2187

=n

∏q=1

m

∏r=1

E(exp(isqr(ψkr

(tq)−ψkr

(tq−1

))))because ψkr

(tq)−ψkr

(tq−1

)is normally distributed with variance tq− tq−1 and mean 0.

By Proposition 59.11.1 on Page 1891, it follows the random variables of 64.3.10 are inde-pendent. Note that as a special case, this also shows the random variables, {ψk (t)}

∞

k=1 areindependent due to the fact ψk (0) = 0.

Recall Corollary 61.11.4 which is stated here for convenience.

Corollary 64.3.2 Let E be any real separable Banach space. Then there exists a sequence,{ek} ⊆ E such that for any {ξ k} a sequence of independent random variables such thatL (ξ k) = N (0,1), it follows

X (ω)≡∞

∑k=1

ξ k (ω)ek

converges a.e. and its law is a Gaussian measure defined on B (E). Furthermore, ||ek||E ≤λ k where ∑k λ k < ∞.

Now let {ψk (t)} be the sequence of Wiener processes described in Lemma 64.3.1.Then define a process with values in E by

W (t)≡∞

∑k=1

ψk (t)ek (64.3.11)

Then ψk (t)/√

t is N (0,1) and so by Corollary 61.11.4 the law of

W (t)/√

t =∞

∑k=1

(ψk (t)/

√t)

ek

is a Gaussian measure. Therefore, the same is true of W (t) . Similar reasoning applies tothe increments, W (t)−W (s) to conclude the law of each of these is Gaussian. Considerthe question whether the increments are independent. Let 0 ≤ t0 < t1 < · · · < tm and letφ j ∈ E ′. Then by the dominated convergence theorem and the properties of the {ψk} ,

E

(exp

(i

m

∑j=1

φ j(W (t j)−W

(t j−1

))))

= E

(exp

(i

m

∑j=1

(∞

∑k=1

(ψk (t j)−ψk

(t j−1

))φ j (ek)

)))

= E

(m

∏j=1

exp

(i

∞

∑k=1

(ψk (t j)−ψk

(t j−1

))φ j (ek)

))