Kenneth Kuttler

64.5. HILBERT SPACE VALUED WIENER PROCESSES 2195

Note the characteristic function of a Q Wiener process is

ei(h,W (t)))= e−

12 t2(Qh,h) (64.5.18)

Note that by Theorem 61.8.5 if you simply say that the distribution measure of W (t)is Gaussian, then it follows there exists a trace class operator Qt and mt ∈ H such that thismeasure is N (mt ,Qt) . Thus for W (t) a Wiener process, Qt = tQ and mt = 0. In addition,the increments are independent so this is much more specific than the earlier definition ofa Gaussian measure.

What is a Q Wiener process if the Hilbert space is Rn? In particular, what is Q? It isgiven that

L ((h,W (t)−W (s))) = N (0,(t− s)(Qh,h))

In this case everything is a vector in Rn and so for h ∈ Rn,

eiλ (h,W (t)−W (s)))= e−

12 λ

2(t−s)(Qh,h)

In particular, letting λ = 1 this shows W (t)−W (s) is normally distributed with covariance(t− s)Q because its characteristic function is e−

12 h∗(t−s)Qh.

With this and definition, one can describe Hilbert space valued Wiener processes in afairly general setting.

Theorem 64.5.2 Let U be a real separable Hilbert space and let J : U0→U be a HilbertSchmidt operator where U0 is a real separable Hilbert space. Then let {gk} be a completeorthonormal basis for U0 and define for t ∈ [0,T ]

W (t)≡∞

∑k=1

ψk (t)Jgk

Then W (t) is a Q Wiener process for Q = JJ∗ as in Definition 64.5.1. Furthermore, thedistribution of W (t)−W (s) is the same as the distribution of W (t− s) , and W is Holdercontinuous with exponent γ for any γ < 1/2. There also is a subsequence denoted by Nsuch that the convergence of the series

∑k=1

ψk (t)Jgk

is uniform for all ω not in some set of measure zero.

Proof: First it is necessary to show the series converges in L2 (Ω;U) for each t. Forconvenience I will consider the series for W (t)−W (s) . (Always, it is assumed t > s.)Then since ψk (t)−ψk (s) is normal with mean 0 and variance (t− s) and ψk (t)−ψk (s)and ψ l (t)−ψ l (s) are independent,

∫Ω

∣∣∣∣∣ n

∑k=m

(ψk (t)−ψk (s))Jgk

∣∣∣∣∣2

=∫

Ω

∑k,l=m

((ψk (t)−ψk (s))Jgk,(ψ l (t)−ψ l (s))Jgl)

64.5. HILBERT SPACE VALUED WIENER PROCESSES 2195Note the characteristic function of a Q Wiener process isE (ei) =e (Qhh) (64.5.18)Note that by Theorem 61.8.5 if you simply say that the distribution measure of W (rt)is Gaussian, then it follows there exists a trace class operator Q, and m, € H such that thismeasure is N (m,,Q,). Thus for W (t) a Wiener process, Q; = tQ and m, = 0. In addition,the increments are independent so this is much more specific than the earlier definition ofa Gaussian measure.What is a Q Wiener process if the Hilbert space is R”? In particular, what is Q? It isgiven thatL ((h,W (t) —W(s))) =N (0, (ts) (Qh,h))In this case everything is a vector in R” and so for h € R”,E (einwir-win)) — p34 (t-8)(Qh,h)In particular, letting A = 1 this shows W (t) — W (s) is normally distributed with covariance: oo. : . 1 px(t — s) Q because its characteristic function is e~ 2’ (~)2",With this and definition, one can describe Hilbert space valued Wiener processes in afairly general setting.Theorem 64.5.2 Let U be a real separable Hilbert space and let J: Uj + U be a HilbertSchmidt operator where Up is a real separable Hilbert space. Then let {g,} be a completeorthonormal basis for Up and define for t € [0,T]WS y(t) Je%k=1Then W (t) is a Q Wiener process for Q = JJ* as in Definition 64.5.1. Furthermore, thedistribution of W (t) — W(s) is the same as the distribution of W (t — s), and W is Holdercontinuous with exponent y for any y < 1/2. There also is a subsequence denoted by Nsuch that the convergence of the seriesNY vy (t) Jeek=1is uniform for all @ not in some set of measure zero.Proof: First it is necessary to show the series converges in L* (Q;U) for each t. Forconvenience I will consider the series for W (t) — W(s). (Always, it is assumed ft > s.)Then since y;, (t) — yw; (s) is normal with mean 0 and variance (ft —s) and wy, (t) — y;, (s)and y; (t) — yw; (s) are independent,L (We) — We (s)) Jax] dP= [Yu vels)) Seu (wi) — wis) Ja0Q k,l=m2n