1.4 Cameron-Martin Space
Proposition 1.4.1 ([Theorem 8.2.6, Str24]).label Let $(H, E, \wien)$ be an abstract Wiener space, then the operators
are compact. In particular, $H \in \cb(E)$.
Proof. It is sufficient to show that both operators map weakly convergent sequences to strongly convergent sequences.
Let $\seq{x_n^*}\subset E^{*}$ such that $x_{n}^{*} \to x \in E^{*}$ weakly, then since $\wh \wien: E^{*} \to \complex$ is continuous with respect to sequential weak* convergence,
as $n \to \infty$. Thus $\iota^{*} x_{n}^{*} \to \iota^{*}x^{*}$ strongly in $H$.
To see the compactness of $\iota$, let $\seq{h_n}\subset H$ such that $h_{n} \to 0$ weakly. By the Uniform Boundedness Principle, there exists $R \ge 0$ such that $\sup_{n \in \nat}\norm{h_n}_{H} \le R$. Let $\seq{x_n^*}\subset E^{*}$ be a weak*-dense subset of $E^{*}$ with $\norm{x_n^*}_{E^*}= 1$ for all $n \in \nat$. Let $\eps > 0$, then by compactness of $\seq{\iota^*x^*_n}\subset H$, there exists $N \in \nat$ such that $\seq{\iota^*x^*_n}\subset \bigcup_{n = 1}^{N} B_{H}(\iota^{*}x^{*}_{n}, \eps)$. In which case,
Given that $h_{n} \to 0$ weakly, $\max_{1 \le k \le N}\angles{h_n, x_k^*}_{E} \to 0$ as $n \to \infty$. As $\eps > 0$ is arbitrary, $\norm{h_n}_{E} \to 0$ as $n \to \infty$.$\square$
Proposition 1.4.2.label Let $(H, E, \wien)$ be an abstract Wiener space. If $E$ is infinite-dimensional, then $\wien(H) = 0$.
Proof. Let $\seq{x_n^*}\subset E^{*}$ such that $\seq{\iota^* x_n^*}$ is an orthonormal basis for $H$. For each $n \in \nat$, let $X_{n}(x) = \dpb{x, x_n^*}{E}$, then $\seq{X_n}$ is a family of independent Gaussian random variables with variance $1$. Thus $\sum_{n \in \nat}X_{n}^{2} = \infty$ almost surely. By (2) of Lemma 1.3.1, $\wien(H) = 0$.$\square$
Theorem 1.4.3.label Let $(H, E, \wien)$ be an abstract Wiener space, then
- (1)
There exists a unique linear isometry $\ci: H \to L^{2}(\wien; \real)$ such that $\ci(\iota^{*}x^{*}) = \dpb{\cdot, x^*}{E}$ for all $x^{*} \in E^{*}$.
- (2)
$\bracs{\ci(h): h \in H}$ is a Gaussian family.
The map in (1) is the Paley-Wiener map.
Proof. Let $x^{*} \in E^{*}$, then $\dpb{\cdot, x^*}{E}$ is a centred Gaussian random variable with variance $\norm{\iota^*x^*}_{H}^{2}$. By (3) of Lemma 1.3.1, $\iota^{*}(E^{*}) \subset H$ is dense. Thus $\ci$ extends uniquely into a continuous linear isometry.$\square$
Theorem 1.4.4 (Cameron-Martin, [Theorem 8.2.7, Str24]).label Let $(H, E, \wien)$ be an abstract Wiener space and $g \in E$. Denote $\tau_{g}: E \to E$ with $x \mapsto x + g$ as the translation map, then:
- (1)
If $g \in H$, then $(\tau_{g})_{*} \ll \wien$ and
\[\frac{d(\tau_{g})_{*}\wien}{d\wien}(x) = e^{\ci (g) - \norm{g}_H^2/2}\] - (2)
If $g \not\in H$, then $(\tau_{g})_{*} \wien \perp \wien$.
Proof. (1): Let $\mu = (\tau_{g})_{*} \wien$, then for any $x^{*} \in E^{*}$,
On the other hand, for any $x, y \in H$, $\ci(x)$ and $\ci(y)$ are jointly Gaussian, so for any $\xi, \eta \in \complex$,
In particular, if $\xi = 1$, $\eta = i$, $x = g$, and $y = x^{*}$, then
(2): Let $x^{*} \in E^{*}$ with $\norm{\iota^*x^*}_{H} = 1$, and $\mathcal{F} = \sigma(x^{*})$, then
If $\mu = \lambda + Fd\wien$ is the Lebesgue decomposition of $\mu$ with respect to $\wien$, then
Given that $\norm{\iota^*x^*}_{H} = 1$,
where $\ev^{\wien}(F^{2})$ does not depend on $x^{*}$. If $F \ne 0$, then the above bound implies that
Thus $g \in H$.$\square$