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As far as I am aware the fractional Brownion motion will not satisfy the Markov property, so even if you can make sense of the diffusion equation (how?), it will not be Markovian. Therefore I would guess it will not be the limit of a Markov chain, as such a limit would be Markovian. Unless, perhaps, you consider limits of 'higher order' Markov chains, e.g. a Markov chain in $\mathbb R^d$, and consider the limit of e.g. the first component.
Of course, nice example, the stationary measures being the constant distribution as well as the one for which $\mu(n+1) = \alpha \mu(n)$ for some $\alpha \neq 1$. Thanks!
