Schilder's theorem from large deviations theory tells us that scaled Brownian motion $\sqrt{\varepsilon} W_t$ on Wiener space $C_0([0,T],\Bbb R^d)$ satisfies a large deviation principle with good rate function:
$$I(\omega)=\begin{cases}\frac{1}{2}\int_0^T |\dot{\omega}(t)|^2~dt &\text{ if } \omega\in \mathcal H\\ \infty &\text{ if } \omega\not\in\mathcal H\end{cases}$$
Where $\mathcal H$ is the Cameron Martin space of Brownian motion (see also this).
This seems to be too coincidental. Why do we only care about the paths that are in the Cameron Martin space? Cameron martin theorem tells us that the law of Brownian motion is quasi invariant under translations by Cameron Martin directions. This theorem seems to tell us that paths outside the CM space decay very quickly in probability.
What is the connection? Is there anyway of understanding Schilder's theorem by CM theorem?
Figured it out.
Let $(\mathcal B, \mu)$ be a separable Banach space $\mathcal B$ with Gaussian measure $\mu$.
Theorem (Cameron-Martin): for $h\in \mathcal B$, define the map $T_h:\mathcal B\to \mathcal B$ by $T_h(x)=x+h$. Then the measure $T_h^\ast\mu$ is absolutely continuous with respect to $\mu$ if and only if $h\in \mathcal H_{\mu}$, the Cameron Martin space.
We have the Radon Nikodym derivative:
$$\frac{dT_h^\ast \mu}{d\mu}=\exp\left( h^\ast(x)-\frac{1}{2}\|h\|_{\mathcal H_\mu}^2\right)$$
In computing the good rate function we have to apply a Girsanov-type change of measure. However this transformation only makes sense if $h \in \mathcal H_{\mu}$. The expession in the $\exp$ is exactly what you take the $\sup$ of to get the Legendre transform (which via Gaertner-Ellis/Cramer is the good rate function).
So (without full details checked, but I believe this should be correct):
Theorem: Given a separable Banach space $\mathcal B$ with centered Gaussian measure $\mu$ and covariance $C_\mu$, then the measures $\{\mu_\epsilon\}$ given by the covariance $C_\mu^\epsilon=\epsilon C_\mu$ satisfy a LDP with good rate function:
$$I(\omega)=\begin{cases}\frac{1}{2}\|\omega\|_{\mathcal H_\mu}^2&\text{ for } \omega \in \mathcal H_\mu\\\infty &\text{ for } \omega\not\in \mathcal H_\mu \end{cases}$$