If we want to make use of our method in the in nite-dimensional case, our assumptions require amongst others that all reference measures are absolutely continuous with respect to each other. This forces us to impose severe restric-tions on the time-dependent drift part of our Ornstein-Uhlenbeck equation. The following proposition gives a feasible choice where the drift is minus identity up to a (time-dependent) trace class operator.
Proposition 7.5.1 In our setting let (en)n be an orthonormal basis of H di-agonalizing the covariance matrix R of our Brownian noise. Let us denote its Eigenvalues by rn. Let then −A(t) be diagonalizable in the same basis with Eigenvalues λn(t) = 1 + fn(t)rn where the real-valued functions fn are smooth and bounded away from zero and in nity, that is we have a positive constant C such that:
• 0 < C1 ≤ fn(t) ≤ C < ∞for all n ∈ N, t ∈ R,
• supn,t
dtdfn(t) ≤ C.
Then Assumption 7.1.4 is ful lled for the reference measures (νt).
Proof Let us denote by Rt the covariance operator of νt. We have Rt = R∞
0 esA(t)ResA∗(t)ds. Since all operators are diagonal in the same basis we obtain the Eigenvalues rn(t) of Rt as one-dimensional integrals, in a simple calculation:
Rten= Z ∞
0
esA(t)ResA∗(t)ends = rn
Z ∞ 0
e2sλn(s)ends = 1 2
rn
1 + fn(t)rn
en.
Thus we have rn(t) = 121+frn
n(t)rn.
For the calculation of the Entropy, note that we have the well-known property for product measures:
Ent⊗N
n=1µn(⊗Nn=1νn) =
N
X
n=1
Entµn(νn).
By Lemma 7.5.3 this is still true in our in nite-dimensional setting so that we have:
Entνs(νt) =
∞
X
n=1
EntN (0,rn(s))(N (0, rn(t))).
So we obtain, using Lemma 7.5.2 for the calculation of the one-dimensional
7.5. EXAMPLE 85
where we have used boundedness and positivity of the fn and the mean value theorem for the logarithm. Thus, Assumption 7.1.4 (i) is ful lled.
In this setting we can also calculate the Radon-Nikodym derivatives dνdνts accord-ing to Lemma 7.5.3.
Denoting xn = hx, eni, we obtain:
where we used the mean value theorem and the uniform bound on the deriva-tives of the functions fn.
Hence, Assumption 7.1.4 (ii) is ful lled, since f(s, t) := Q∞n=1 rn(s)
rn(t) is multi-plicative as required and convergence of the in nite product is seen easily.
Assumption 7.1.4 (iii) follows easily since the relevant operators commute and Assumption 7.1.4 (iv) holds since we have rn(t) ≤ rn for all n ∈ N. Lemma 7.5.2 Let ν1 := νΣ1 and ν2 := νΣ2 be centered Gaussian measures on Rn with covariance matrices Σ1 and Σ2 respectively. Let µ be any measure absolutely continuous with respect to Lebesgue measure and admitting a second moment. Then we have
(i) Entν1(ν2) =1
86 CHAPTER 7. A VARIATIONAL INTERPRETATION and the result follows along the preceding calculations.
Lemma 7.5.3 (Radon-Nikodym derivative) For i = 1, 2, let µi= N (0, Σi) be Gaussian measures on H with covariance operators Σi. We assume that both Σ1 and Σ2 have the same eigenbasis (en)n∈N and we denote the eigenvectors by σni respectively. Set µin := N (0, σni), a normal distribution on R, so that we have formally: µi = ⊗n∈Nµin. If µ1 and µ2 are equivalent, then we can write the Radon-Nikodym derivative as a tensor product:
dµ1
dµ2(x) = Y
n∈N
dµ1n
dµ2n(hx, eni) x ∈ H Proof Let us denote
µ0(dx) := Y
n∈N
dµ1n
dµ2n(hx, eni)µ2(dx), then we have to show that
Z
holds for any measurable and bounded function F . By a monotone class argu-ment it is su cient to show it for functions F of the form
F (x) = f (hx, e1i, ..., hx, eNi), (∗)
7.5. EXAMPLE 87
where N ∈ N is arbitrary and f : RN → R is a smooth function. The sys-tem formed by these functions is multiplicative and since its countable subset (hx, eNi)N ∈N separates the points of H the system also generates σ(H).
Thus, let us take a function F as in (∗) and check (7.9):
Z
H
F (x)µ1(dx) = Z
H
f (hx, e1i, ..., hx, eNi)µ1(dx)
= Z
RN
f (x) (P]Nµ1) (dx)
= Z
RN
f (x) (⊗Nn=1µ1n) (dx)
= Z
RN
f (x1, ..., xN)
N
Y
n=1
dµ1n dµ2n(xn)
µ21(dx1)...µ2N(dxN)
= Z
RN
f (x) (P]Nµ0) (dx) = Z
H
F (x)µ0(dx),
where PN : H → RN, x 7→ (hx, e1i, ..., hx, eNi).
88 CHAPTER 7. A VARIATIONAL INTERPRETATION
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