= Brownian change of measure by a positive space-time harmonic function
{c}
{title2=$dX_t=dB_t^{\mathbb Q}+\partial_x\log h(X_t,t)dt$}
If $h>0$ is smooth and $Z_t=h(X_t,t)$ is the unit-mean density martingale of a locally absolutely continuous probability measure relative to Wiener measure, then $h_t+\frac12h_{xx}=0$. The <Itô formula> gives $Z=\mathcal E(\int(h_x/h)(X_s,s)dX_s)$ when $Z_0=1$. The true-martingale hypothesis is already supplied by the density process. On each finite horizon the <Girsanov theorem> makes $X-\int(h_x/h)(X_s,s)ds$ Brownian under the new measure. Positivity provides local equivalence; global infinite-horizon equivalence and the Novikov condition are not additional requirements.
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