Cameron-Martin theorem

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.

If $h(t)={\int }_0^t{g}_{s}ds$ with deterministic $g\in L^2(0,\infty )$, the finite-time Cameron-Martin theorem densities have second moments $e^{{\int }_0^t{g}_s^{2}ds}$ bounded uniformly in time. Their uniformly integrable martingale limit is the strictly positive displayed density, giving equivalence on the full infinite continuous-path space. Finite-time densities alone do not guarantee such a global density: a nonzero constant drift has infinite energy and its infinite-horizon path law is singular, distinguished by the limiting ratio $B_t/t$.

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

For ${\tau }_{a,b}=\inf \{t\ge 0:B_t+bt=a\}$ with $a,b>0$,The Cameron-Martin likelihood at a driftless path hitting $a$ at time $t$ is $e^{ab-b^{2}t/2}$.