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One usually deals with a discrete time Markov process in the following form: given a state space $E$ the Markov process is defined by transition kernel $T(B|x)$ such that $ \mathsf{P}(X_1\in B|X_0 = x) = T(B|x) $ for all $x\in E$, $B\in\mathcal{B}(E)$.

So, given a current state we have a distribution of the future state.

On the other hand it can be an interesting problem given a current state to find a distribution of the previous state which "fits" with a transition kernel $T$.

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    So? You might care to correct this paragraph in your post...2011-04-25

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This is called time-reversal of Markov chains and is presented in numerous lecture notes available on the web, see here for example.

Remark As said in the comments, the sentence of the post stating that given a state $X_{n+1}$ we know the distribution of $X_n$ signals a deep misunderstanding of the structure of Markov processes. In the example considered, $v_n$ is not independent on $X_{n+1}$ and one should note that, given any nonzero $a$ and $b$ and any random $X$, any random variable $Y$ can be written as $Y=X/a-(b/a)V$ for a well chosen random variable $V$.