= Solution
For $0\leq s\leq t$ and $E\in\mathcal F_s$, the definition of the <Radon-Nikodym derivative> gives
$$
\mathbb E_{\mathbb P}[\mathbf1_E Z_t]=\mathbb Q(E)=\mathbb E_{\mathbb P}[\mathbf1_E Z_s].
$$
The density is nonnegative and integrable; with the probability normalization used here its expectation is one. Hence
$$
\boxed{\mathbb E_{\mathbb P}[Z_t\mid\mathcal F_s]=Z_s,}
$$
so $Z$ is a <martingale>. When $Z_t=h(X_t,t)$, the <Itô formula> under Wiener measure gives
$$
dZ_t=h_x(X_t,t)dX_t+g(X_t,t)dt,\qquad g=h_t+\frac12h_{xx}.
$$
The 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 $\int_0^tg(X_s,s)ds=0$ 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 $x$. Therefore
$$
\boxed{h_t(x,t)+\frac12h_{xx}(x,t)=0\qquad(x\in\mathbb R,\ t>0).}
$$
If 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.
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