Please imagine a discrete random walk on a one-dimensional lattice. The lattice consists of a set of $L$ positions, $(x_0, x_1, ..., x_L) \in L$, where $x_0$ is the initial position of the walk (as well as a reflecting boundary), and $x_L$ is absorbing.

For each position in the walk, one of $N$ jump probabilities ($N \leq L$) is assigned (forward - $p_k$, backward - $(1-p_k)$) from a set $P$, where $(p_1, p_2, ..., p_N) \in P$. However, we do not have knowledge about these assignments. All we are provided with is a set $M$, $(m_1, m_2, ..., m_L) \in M$, of mean occupancy values for each position in the one-dimensional lattice, $(x_0, x_1, ..., x_L) \in L$.

Now, provided access to $M$, to what extent can we find the values for the set of jump probabilities, $(p_1, p_2, ..., p_N) \in P$ (as defined above), for each position in the lattice, $x_k$? Can we guarantee a unique solution by placing certain restrictions on the finite set of jump probabilities $P$?

(Note - This is the reverse formulation of an earlier question I asked about computing mean occupancy for sites in the one-dimensional random walk from assigned jump probabilities. See below for the earlier question.)

Please imagine a discrete random walk on a one-dimensional lattice. The lattice consists of a set of $L$ positions, $(x_0, x_1, ..., x_L) \in L$, where $x_0$ is the initial position of the walk (as well as a reflecting boundary), and $x_L$ is absorbing.

For each position in the walk, we assign one of $N$ jump probabilities (forward, $p_k$, backward, $(1-p_k)$) from a set $P$, where $(p_1, p_2, ..., p_N) \in P$.

For the duration of the random walk, until the absorbing target $x_L$ is reached, what is the mean occupancy of the a given position in the one-dimensional lattice, $x_k$? I'm hoping to find an efficient method to compute an exact solution.