Brownian change of measure by a positive space-time harmonic function ($d{X}_t=d{B}_t^{\mathbb{Q}}+{\partial }_x\log h(X_t,t)dt$)

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 $h_t+\frac{1}{2}{h}_{xx}=0$. The Itô formula gives $Z=\mathcal{E}(\int (h_x/h)(X_s,s)d{X}_s)$ 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.