64.5. HILBERT SPACE VALUED WIENER PROCESSES 2197

= limN→∞

exp

(N

∑j=1−1

2r2 (h,Jg j)

2 (t− s)

)

= exp

(−1

2r2 (t− s)

∞

∑j=1

(h,Jg j)2U

)

= exp

(−1

2r2 (t− s)

∞

∑j=1

(J∗h,g j)2U0

)

= exp(−1

2r2 (t− s) ||J∗h||2U0

)= exp

(−1

2r2 (t− s)(JJ∗h,h)U

)= exp

(−1

2r2 (t− s)(Qh,h)U

)(64.5.19)

which shows (h,W (t)−W (s))U is normally distributed with mean 0 and variance

(t− s)(Qh,h)

where Q ≡ JJ∗. It is obvious from the definition that W (0) = 0. Note that Q is of traceclass because if {ek} is an orthonormal basis for U,

∑k(Qek,ek)U = ∑

k||J∗ek||2U0

= ∑k

∑l(J∗ek,gl)

2U0

= ∑l

∑k(ek,Jgl)

2U = ∑

l||Jgl ||2U < ∞

To find the covariance, consider

E ((h1,W (t)−W (s))(h2,W (t)−W (s))) ,

This equals

E

(∞

∑k=1

(ψk (t)−ψk (s))(h1,Jgk)∞

∑j=1

(ψ j (t)−ψ j (s)

)(h2,Jg j)

).

Since the series converge in L2 (Ω;U) , the independence of the ψk (t)−ψk (s) implies theabove equals

= limn→∞

E

(n

∑k=1

(ψk (t)−ψk (s))(h1,Jgk)n

∑j=1

(ψ j (t)−ψ j (s)

)(h2,Jg j)

)

= limn→∞

(t− s)n

∑k=1

(h1,Jgk)(h2,Jgk)

= limn→∞

(t− s)n

∑k=1

(J∗h1,gk)U0(J∗h2,gk)U0

= (t− s)∞

∑k=1

(J∗h1,gk)U0(J∗h2,gk)U0

= (t− s)(J∗h1,J∗h2) = (t− s)(Qh1,h2) .