Finding jump probabilities from mean-occupancy values for positions on a one-dimensional random walk

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.

• Is the measure on $P$ uniform? – Steve Huntsman Jun 3 '10 at 20:00
• If so, consider a given assignment of elements of $P$. The concomitant transition matrix can be constructed straightforwardly (including a "coffin state") and the associated fundamental matrix as well ( books.google.com/… ). This will give you the information you need for that particular assignment. Then average over the assignments from $P$. – Steve Huntsman Jun 3 '10 at 20:06
• Steve, thank you, I appreciate the link. I suppose my question though was - given a 'particular' random assignment of jump probabilities, can one do better than averaging over all the assignments from P to find the mean occupancy for a position $x_k$? Aside from the contribution of a particular initialization state, my intuition was that the mean occupancy would not be the same for all sites. – Rob Grey Jun 3 '10 at 21:31
• Dear Steve, I updated my question to reflect my previous comment. Perhaps it will make a difference? My apologies for that. – Rob Grey Jun 3 '10 at 21:34
• If you have occupancy times of a birth n' death process until hitting some particular state they are geometric, and if you put the process in continuous time they are exponential, and I think you can relate even joint occupancies to values of bessel processes via imbedding them in brownian motion & using something like a Ray-Knight theorem. – mike May 11 '12 at 14:07