64.5. HILBERT SPACE VALUED WIENER PROCESSES 2205

which is the same as E (exp i(h,W (t− s))U ) due to the fact W (0) = 0. This proves thetheorem.

The above shows there exists Q Wiener processes in any separable Hilbert space. NextI will show the way described above is the only way it can happen.

Theorem 64.5.4 Suppose {W (t)} is a Q Wiener process in U, a real separable Hilbertspace. Then letting

Q =∞

∑k=1

λ kek⊗ ek

where the {ek} are orthonormal, λ k ≥ 0, and ∑∞k=1 λ k < ∞, it follows

W (t) =∞

∑k=1

√λ kψk (t)ek (64.5.28)

where

ψk (t)≡

{1√λ k

(W (t) ,ek)U if λ k ̸= 0

0 if λ k = 0

then {ψk (t)} is a Wiener process and for t0 < t1 < · · ·< tn the random variables{ψk (tq)−ψk

(tq−1

): (q,k) ∈ (1,2, · · · ,n)× (k1, · · · ,km)

}are independent. Furthermore, the sum in 64.5.28 converges uniformly for a.e. ω on anyclosed interval, [0,T ].

Proof: First of all, the fact that W (t) has values in U and that {ek} is an orthonormalbasis implies the sum in 64.5.28 converges for each ω. Consider

E (exp(irψk (t))) = E

(exp

(ir

1√λ k

(W (t) ,ek)U

))Since W (t) is given to be a Q Wiener process, (W (t) ,ek)U is normally distributed withmean 0 and variance t (Qh,h) . Therefore, the above equals

= e− 1

2 r2 1λk

t(Qek,ek) = e− 1

2 r2 1λk

tλ k = e−12 r2t ,

the characteristic function for a random variable which is N (0, t) . The independence of theincrements for a given ψk (t) follows right away from the independence of the incrementsof W (t) and the distribution of the increments being N (0,(t− s)) follows similarly to theabove.

For t1 < t2 < · · ·< tn, why are the random variables,{(W (tq) ,ek)U −

(W(tq−1

),ek)

U : (q,k) ∈ (1,2, · · · ,n)× (k1, · · · ,km)}

(64.5.29)

independent? Let

P =n

∑q=1

m

∑j=1

sq j

((W (tq) ,ek j

)U−(

W(tq−1

),ek j

)U

)