= Schilder theorem
{c}
{title2=$I(\phi)=\frac12\int|\dot\phi|^2$}
The Schilder theorem gives a <large deviation principle> for small <Brownian motion> paths. For standard Brownian motion $W$ starting at zero on $[0,T]$, the laws of $\sqrt\varepsilon W$ on continuous paths with the uniform <norm> have speed $1/\varepsilon$ and <good rate function>
$$
I(\phi)=\begin{cases}\frac12\int_0^T|\dot\phi(t)|^2\,dt,&\phi(0)=0,\ \phi\text{ absolutely continuous},\ \dot\phi\in L^2,\\+\infty,&\text{otherwise}.\end{cases}
$$
In particular, the zero starting value and finite-energy path class are part of the rate. Typical <Brownian motion> paths are not themselves absolutely continuous; the rate describes the exponential cost of neighborhoods of prescribed paths.<An Introduction to Large Deviations by Vitalii Konarovskyi>[Section7.1, Theorem7.1, and section7.3 proof]{ref}
For paths constrained by $\phi(T)=b$, the <Cauchy-Schwarz inequality> gives $\int_0^T|\dot\phi|^2dt\geq b^2/T$, with equality for $\phi(t)=bt/T$. Thus the least endpoint cost is $b^2/(2T)$, agreeing with the exponential scale of the <normal distribution> of $\sqrt\varepsilon W_T$.
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