Markov Chain w/ non-stochastic matrix

Question

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 M^T (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.