64.6. WIENER PROCESSES, ANOTHER APPROACH 2221

for any h ∈H that the random variables (W (tk)−W (tk−1) ,h)H are independent, it followsthat the random variables W (tk)−W (tk−1) are also.

What of the Holder continuity? In the above computation for independence, as a specialcase, for λ ∈ H,

E (exp i(λ ,W (t)−W (s))) = exp(−1

2|J∗λ |2U (t− s)

)(64.6.40)

In particular, replacing λ with λ r for r real,

E (exp ir (λ ,W (t)−W (s))) = exp(−1

2r2 |J∗λ |2U (t− s)

)Now we differentiate with respect to r and then take r = 0 as before to obtain finally that

E((λ ,W (t)−W (s))2m

)≤Cm |J∗λ |2m |t− s|m =Cm (Qλ ,λ )m |t− s|m

Then letting {hk} be an orthonormal basis for H, and using the above inequality withMinkowski’s inequalitiy,

(E(|W (t)−W (s)|2m

))1/m=

(E

([∞

∑k=1

(W (t)−W (s) ,hk)2

]m))1/m

≤∞

∑k=1

[E((W (t)−W (s) ,hk)

2m)]1/m

≤∞

∑k=1

(Cm (t− s)m |J∗hk|2m

U

)1/m

= C1/mm |t− s|

∞

∑k=1|J∗hk|2U =C1/m

m |t− s|∞

∑k=1

∞

∑j=1

(J∗hk,g j)2

= C1/mm |t− s|

∞

∑j=1

∞

∑k=1

(hk,Jg j)2 = |t− s|C1/m

m

∞

∑j=1

∣∣Jg j∣∣2H

and since J is Hilbert Schmidt, modifying the constant yields

E(|W (t)−W (s)|2m

)≤Cm |t− s|m

By the Kolmogorov Centsov theorem, Theorem 62.2.3,

E(

sup0≤s<t≤T

∥W (t)−W (s)∥(t− s)γ

)≤Cm

whenever γ < β/α = m−12m . Thus the above is true whenever γ < 1/2. Hence off a set of

measure zero, t→W (t) is Holder continuous.What of the covariance condition? From 64.6.40, letting f ,g be two elements of H,

E (exp i(α f +βg,W (t)−W (s))) = exp(−1

2(Q(α f +βg) ,α f +βg)(t− s)

)