Let $(\nu_t)_{t \in [0,1]}$ be Borel probability measures on a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t \in [0,1]})_t,\mathbb{P})$ and suppose that $(X_t)_{t \in [0,1]}$ is a stochastic process for which $X_t \sim \nu_t$ for every $t \in [0,1]$.

Can we identify if any such $X_{\cdot}$ is a semi-martingale by looking only at measures $\nu_{\cdot}$?

Let $(\nu_t)_{t \in [0,1]}$ be Borel probability measures on a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t \in [0,1]})_t,\mathbb{P})$.

Does there exist a semi-martingale $(X_t)_{t\in[0,1]}$ on this stochastic basis, such that $X_t\sim \nu_t$ for everty $t \in [0,1]$? If not, what conditions are needed on the measures $\nu_{\cdot}$ for this to be possible?