64.5. HILBERT SPACE VALUED WIENER PROCESSES 2199

N→ ∞, this implies

E

(exp

m

∑j=1

i(h j,W (t j)−W

(t j−1

))U

)

=m

∏j=1

exp

(−1

2

∞

∑k=1

(h j,Jgk)2U

(t j− t j−1

))

=m

∏j=1

exp

(−1

2(t j− t j−1

) ∞

∑k=1

(J∗h j,gk)2U0

)

=m

∏j=1

exp(−1

2(t j− t j−1

)∣∣∣∣J∗h j∣∣∣∣2

U0

)=

m

∏j=1

exp(−1

2(t j− t j−1

)(Qh j,h j)U

)=

m

∏j=1

exp(i(h j,W (t j)−W

(t j−1

))U

)(64.5.20)

from 64.5.19, letting r = 1. By Theorem 59.13.3 on Page 1898, this shows the incrementsare independent.

It remains to verify the Holder continuity. Recall

W (t) =∞

∑k=1

Jgkψk (t)

where ψk is a real Wiener process.Next consider the claim about Holder continuity. It was shown above that

E (exp(ir (h,(W (t)−W (s)))U )) = exp(−1

2r2 (t− s)(Qh,h)U

)Therefore, taking a derivative with respect to r two times yields

E((−(h,(W (t)−W (s)))2

U

)exp(ir (h,(W (t)−W (s)))U )

)= −(t− s)(Qh,h)exp

(−1

2r2 (t− s)(Qh,h)U

)+

r2 (t− s)2 (Qh,h)2U exp

(−1

2r2 (t− s)(Qh,h)U

)Now plug in r = 0 to obtain

E((h,(W (t)−W (s)))2

U

)= (t− s)(Qh,h) .

Similarly, taking 4 derivatives, it follows that an expression of the following form holds.

E((h,(W (t)−W (s)))4

U

)=C2 (Qh,h)2 (t− s)2 ,