2202 CHAPTER 64. WIENER PROCESSES

and this converges to 0 as m,n→ ∞ because it was given that

∞

∑j=1

λ j < ∞

so the series in 64.5.22 converges in L2 (Ω;U) .Therefore, there exists a subsequence{

N

∑k=1

√λ kek (ψk (t)−ψk (s))

}

which converges pointwise a.e. to W (t)−W (s) as well as in L2 (Ω;U) as N → ∞. Thenletting h ∈U,

(h,W (t)−W (s))U =∞

∑k=1

√λ k (ψk (t)−ψk (s))(h,ek) (64.5.23)

Then by the dominated convergence theorem,

E (exp(ir (h,(W (t)−W (s)))U ))

= limN→∞

E

(exp

(ir

(N

∑j=1

√λ j

(ψ j (t)−ψ j (s)

)(h,e j)

)))

= limN→∞

E

(N

∏j=1

eir√

λ j(ψ j(t)−ψ j(s))(h,e j)

)Since the random variables ψ j (t)−ψ j (s) are independent,

= limN→∞

N

∏j=1

E(

eir√

λ j(ψ j(t)−ψ j(s))(h,e j))

Since ψ j (t) is a real Wiener process,

= limN→∞

N

∏j=1

e−12 r2λ j(t−s)(h,e j)

2

= limN→∞

exp

(N

∑j=1−1

2r2

λ j (t− s)(h,e j)2

)

= exp

(−1

2r2 (t− s)

∞

∑j=1

λ j (h,e j)2

)

= exp(−1

2r2 (t− s)(Qh,h)

)(64.5.24)