Let $(\Omega\_1, \mathcal{F}\_1, P\_1)$ and $(\Omega\_2, \mathcal{F}\_2, P\_2)$ be probability spaces and suppose $(X\_t)$ and $(Y\_t)$ are real-valued stochastic processes defined on the respective spaces. Furthermore, assume $(Y\_t)$ is a Markov process. My question is the following:


Suppose $P\_1\circ X\_t^{-1}=P\_2\circ Y\_t^{-1}$ as measures on $\mathbb{R}$. Must $(X\_t)$ be a Markov process? 



Stochastic processes having Markov kernels
No, that both processes have the same one-dimensional marginals is not sufficient. In contrary, when $X$ is an arbitrary elliptic Itô-process, you can always find a Markov process with the same marginals. Cf. I. Gyöngy, Mimicking the One-Dimensional Marginal Distributions of Processes Having an Itô Differential. Probab.Theory Relat.Fields 71(4), 501–516 (1986)


