Let $X_{\cdot}$ be a $1$-dimensional diffusion process. If I know the value of the
$$\big\{\mathbb{E}[\max\{X_t,c\}\big| X_0 =x\big]:\, c\in \mathbb{R} \text{ and } \,\, t\in (0,1] \big\}.$$
Then, can you reconstruct the law of $X_{\cdot}$ on $C([0,1],\mathbb{R})$ or, at least, the marginal laws $\{\mathcal{L}(X_t)\,:\,{t\in [0,1]}\}$?
For any random variable $X$ with $E\max(X,0)<\infty$, you can determine the distribution of $X$ if you know the values of $$g(c):=E\max(X,c)$$ for all real $c$.
Indeed, take any real $c$ and any real $h>0$. Then
$$\begin{aligned}
\frac{g(c+h)-g(c)}h&=E1(X\le c)+E1(c<X\le c+h)\frac{c+h-X}h.
\end{aligned}$$
Noting now that $0\le1(c<X\le c+h)\frac{c+h-X}h\le1(c<X\le c+h)$ and $P(c<X\le c+h)\to0$ as $h\downarrow0$, we conclude that
$$P(X\le c)=g'_+(c),$$
where $g'_+(c)$ is the right derivative of the function $g$ at $c$. $\quad\Box$
