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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!

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See here: mathoverflow.net/questions/78397/… –  Byron Schmuland Feb 2 '13 at 23:30
    
Thanks, @ByronSchmuland! –  Tim Feb 3 '13 at 0:16

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