= Solution
Define $k(x,t)=h_x(x,t)/h(x,t)=\partial_x\log h(x,t)$. Positivity and smoothness of $h$ make $k$ continuous and bounded on compact space-time sets. Brownian paths remain in compact spatial sets on each finite horizon, so $\int_0^Tk(X_s,s)^2ds<\infty$ almost surely for every $T$. Thus
$$
M_t=\int_0^tk(X_s,s)dX_s
$$
is a well-defined <continuous local martingale> with $M_0=0$. The backward heat equation from part (b) removes the drift in Itô's formula and yields
$$
dZ_t=h_x(X_t,t)dX_t=Z_t\,dM_t.
$$
By the allowed fact $Z_0=1$ and the uniqueness in part (a),
$$
\boxed{Z_t=\mathcal E(M)_t=\exp\!\left(\int_0^tk(X_s,s)dX_s-\frac12\int_0^tk(X_s,s)^2ds\right).}
$$
The finite-horizon form of the <Girsanov theorem> used here says: if $M_0=0$ is a <continuous local martingale> and its stochastic exponential $Z$ is a strictly positive true <martingale> of mean one on $[0,T]$, then under $d\mathbb Q|_{\mathcal F_T}=Z_Td\mathbb P|_{\mathcal F_T}$ every continuous $\mathbb P$-local <martingale> $N$ becomes the $\mathbb Q$-local <martingale> $N-[N,M]$. <Quadratic variation> is unchanged by this locally equivalent change of measure.
All the assumptions hold. The integrand defining $M$ is locally square integrable as checked above; $Z$ is strictly positive; and it is already a true <martingale> because it is the density process. On each finite horizon $Z_t=\mathbb E[Z_T\mid\mathcal F_t]$, so it is <uniformly integrable> on that horizon. There is no need to establish a separate Novikov condition, nor to assume equivalence on the infinite-time sigma-algebra.
Take $N=X$. Its covariation with $M$ is $[X,M]_t=\int_0^tk(X_s,s)ds$. Hence $B_t=X_t-\int_0^tk(X_s,s)ds$ is a continuous $\mathbb Q$-local <martingale> starting at zero with $[B]_t=t$. The <Lévy characterization of Brownian motion> makes it Brownian under $\mathbb Q$. Since this argument holds on every finite horizon for the same process $B$,
$$
\boxed{X_t=B_t+\int_0^t\frac{h_x(X_s,s)}{h(X_s,s)}ds,\qquad B\text{ is Brownian motion under }\mathbb Q.}
$$
This is the <Brownian change of measure by a positive space-time harmonic function>, a <Doob h-transform> with drift $\partial_x\log h$.
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