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I've come across a problem which at first appeared to be a markov process however the transition matrix of the graph is non-stochastic. That is, the probabilities among edges leaving a node do not sum to 1. However,the probabilities on edges entering the node do sum to 1. Therefore the transpose of the transition matrix IS stochastic.

I've built up a model in excel and noticed a few properties of this "transpose markovian process":

  1. all nodes reach the same steady state value.
  2. the steady state *is* dependent on the initial state (unlike a markov process).
  3. the steady state value can be computed as S0 x MT (the initial state vector times the transpose of the markovian steady state vector... obtained by transposing the transition matrix and solving as a markov process).

My question is this: Is there a name for such a process? In searching for it I've found it to be beyond obscure, so my hope is some well-informed human can catalyze me with some search terms.

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  • $\begingroup$ This is a nice question, but also a pure mathematics question. Should we migrate to Mathematics? $\endgroup$ – Raphael Mar 16 '15 at 7:31
  • $\begingroup$ "Is there a name for X?" is typically not an ideal question. If the answer is "No", how do you plan to evaluate the answer? Instead, I suggest that you first figure out why you are hoping to find a name (what will you do it? what is the real problem you are trying to solve?), and then ask about that -- ask about the real problem that you are trying to solve. Even if someone told you a name, that'd be only a means to an end -- so what's the end that you're striving for? If you're looking for literature about this processes, what questions specifically are you hoping it would answer? $\endgroup$ – D.W. Mar 16 '15 at 7:32
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No, there's no standard name for a process where the transpose of the transition matrix is stochastic (at least, to my knowledge).

However, this process is the time-reversal of a Markov process. That's a remarkable and descriptive property that just about completely characterizes this process. In particular, it seems likely that many things we'd want to know about such a process can be derived from the fact that it's the time-reversal of a Markov process, plus known facts about Markov processes.

Anyway, if you need to describe the process, "time-reversal of a Markov process process" is probably about as good as you're going to get.

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