Wiener measure

Wiener measure is the probability distribution of Brownian motion on a continuous-path space, usually $C_0([0,T])$ or $C_0([0,\infty ))$.

On $C_0([0,T])$, the Cameron-Martin space consists of the absolutely continuous functions $h$ with $h(0)=0$ and $h^{\prime }\in L^2[0,T]$. Its norm is

Translation of Wiener measure by a path $h$ is equivalent to Wiener measure exactly when $h$ belongs to the Cameron-Martin space of Wiener measure. In that case the density isTranslation by any continuous path outside that space produces a singular measure.

For the Cameron-Martin space of Wiener measure path $h(s)=\mu s$, the Cameron-Martin theorem says that translation of Brownian motion by this linear function changes its law through time $t$ by the density