Suppose a stochastic process is Markovian. Let $L$ be its law on its sample path space (note that here I assume its initial distribution is known, not just conditional distributions). If there is another stochastic process with the same law $L$, will it be also Markovian? In other words, is being Markovian completely determined by the law of a stochastic process?
Similar question for Martingale. Is being Martingale completely determined by the law of a stochastic process?
Is it possible to point out what kinds of properties of a stochastic process are determined by its law completely, and what cannot? Note that I have asked if sample path being continuous a.e. is determined completely by the law here.
Thanks and regards!