64.1. REAL WIENER PROCESSES 2183

For example, ignoring the limit questions and proceding formally,

E (exp(iλ (W (t)−W (s)))) = E

(exp

(iλ

(∞

∑k=1

(X(s,t),gk

)L2 ξ k

)))

= E

(∞

∏k=1

exp(

iλ(X(s,t),gk

)L2 ξ k

))

=∞

∏k=1

E(

exp(

iλ(X(s,t),gk

)L2 ξ k

))=

∞

∏k=1

e−12 λ

2(X(s,t),gk)2L2

= exp

(−1

2λ

2∞

∑k=1

(X(s,t),gk

)2L2

)

= exp(−1

2λ

2 (t− s))

which is the characteristic function of a random variable having mean 0 and variance t− s.Finally note the distribution of W (t− s) is the same as the distribution of

W (1)(t− s)1/2 =∞

∑k=1

(X(0,1),gk

)L2 ξ k (t− s)1/2

because the characteristic function of this last random variable is the same as the charac-teristic function of W (t− s) which is e−

12 λ

2(t−s) which follows from a simple computation.Since W (1) is a normally distrubuted random variable with mean 0 and variance 1,

E(

exp(

iλW (1)(t− s)1/2))

= e−12 λ

2(t−s)

which is the same as the characteristic function of W (t− s).Hence for any positive α,

E(|W (t)−W (s)|α

)= E

(|W (t− s)|α

)= E

(∣∣∣(t− s)1/2 W (1)∣∣∣α)

= |t− s|α/2 E(|W (1)|α

)(64.1.3)

It follows from Theorem 62.2.2 that W (t) is Holder continuous with exponent γ where γ isany positive number less than β/α where α/2 = 1+β . Thus γ is any constant less than

α

2 −1α

=12

α−2α

Thus γ is any constant less than 12 .

The proof of the theorem, which only depended on {ξ i}∞

i=1 being independent randomvariables each normal with mean 0 and variance 1, implies the following corollary.