 Schilder theorem

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 $\sqrt{\epsilon }W$ on continuous paths with the uniform norm have speed $1/\epsilon$ and good rate functionIn 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.^[ref]

For paths constrained by $\varphi (T)=b$, the Cauchy-Schwarz inequality gives ${\int }_0^T|\dot{\varphi }{|}^{2}dt\ge b^2/T$, with equality for $\varphi (t)=bt/T$. Thus the least endpoint cost is $b^2/(2T)$, agreeing with the exponential scale of the normal distribution of $\sqrt{\epsilon }{W}_T$.