 Solution

For $0\le s\le t$ and $E\in {\mathcal{F}}_s$, the definition of the Radon-Nikodym derivative givesThe density is nonnegative and integrable; with the probability normalization used here its expectation is one. Henceso $Z$ is a martingale. When $Z_t=h(X_t,t)$, the Itô formula under Wiener measure givesThe first term is a continuous local martingale, by localization on compact space-time sets. Since $Z$ is a martingale, uniqueness of the continuous semimartingale decomposition makes ${\int }_0^{t}g(X_s,s)ds=0$ identically. The integrand is continuous in time along every continuous path, so $g(X_t,t)=0$ for every $t$, outside one common null event.

For each fixed $t>0$, $X_t$ has a Gaussian distribution with strictly positive density everywhere on $\mathbb{R}$. Since $g(\cdot ,t)$ is continuous, $g(X_t,t)=0$ almost surely implies $g(x,t)=0$ for every real $x$. ThereforeIf the stated smoothness includes time zero, continuity extends the equation there. This is the backward heat equation. The full-support argument is needed to pass from an identity along Brownian paths to an equation at every spatial point.