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 $T$.