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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?

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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)

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