Multiplicatively closed Markov models must form Lie algebras

@article{Sumner2017MultiplicativelyCM,
  title={Multiplicatively closed Markov models must form Lie algebras},
  author={Jeremy G. Sumner},
  journal={Anziam Journal},
  year={2017},
  volume={59},
  pages={240-246}
}
  • J. Sumner
  • Published 4 April 2017
  • Mathematics
  • Anziam Journal
We prove that the probability substitution matrices obtained from a continuous-time Markov chain form a multiplicatively closed set if and only if the rate matrices associated with the chain form a linear space spanning a Lie algebra. The key original contribution we make is to overcome an obstruction, due to the presence of inequalities that are unavoidable in the probabilistic application, which prevents free manipulation of terms in the Baker–Campbell–Haursdorff formula. 

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