Theorem 1.4.4: Cameron-Martin, {{\cite[Theorem 8.2.7]{stroock}}}

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 $x^{*} \in E^{*}$,

\[\wh \mu(x^{*}) = \ev^{\wien}\braks{e^{i\angles{x + g, x^*}_E}}= e^{i\dpb{g, x^*}{E} - \norm{\iota^*x^*}_H^2/2}\]

On the other hand, for any $x, y \in H$, $\ci(x)$ and $\ci(y)$ are jointly Gaussian, so for any $\xi, \eta \in \complex$,

\[\ev^{\wien}\braks{e^{\xi \ci(x) + \eta \ci(y)}}= \exp\braks{\frac{1}{2}\paren{\xi^2\norm{x}_H^2 + 2\xi \eta \dpb{x, y}{H} + \eta^2\norm{y}_H^2}}\]

In particular, if $\xi = 1$, $\eta = i$, $x = g$, and $y = x^{*}$, then

\[\ev^{\wien}\braks{e^{\ci (g) + ix^*}}= e^{i\dpb{g, x^*}{E} - \norm{\iota^*x^*}_H^2/2}= \wh \mu(x^{*})\]

(2): Let $x^{*} \in E^{*}$ with $\norm{\iota^*x^*}_{H} = 1$, and $\mathcal{F} = \sigma(x^{*})$, then

\[\frac{d\mu|_{\cf}}{d\wien_{\cf}}(x) = e^{\dpb{g, x^*}{E}\dpb{x, x^*}{E} - \dpb{g, x^*}{E}^2/2}\]

If $\mu = \lambda + Fd\wien$ is the Lebesgue decomposition of $\mu$ with respect to $\wien$, then

\[\frac{d\mu|_{\cf}}{d\wien_{\cf}}\ge \ev^{\wien}[F|\cf]\]

Given that $\norm{\iota^*x^*}_{H} = 1$,

\[e^{\dpb{g, x^*}{E}^2}= \ev^{\wien}\braks{\paren{\frac{d\mu|_{\cf}}{d\wien_{\cf}}}^2}\ge \ev^{\wien}(F^{2})\]

where $\ev^{\wien}(F^{2})$ does not depend on $x^{*}$. If $F \ne 0$, then the above bound implies that

\[\sup_{\substack{x^* \in E^* \\ ||\iota^* x^*||_H = 1}}|\dpb{g, x^*}{E}^{2}| < \infty\]

Thus $g \in H$.$\square$

Brownian Motion and Stochastic Calculus