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Coarse-graining open Markov processes

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John C. Baez and Kenny Courser

Coarse-graining is a standard method of extracting a simpler Markov
process from a more complicated one by identifying states. Here we
extend coarse-graining to `open' Markov processes: that is, those
where probability can flow in or out of certain states called `inputs'
and `outputs'. One can build up an ordinary Markov process from
smaller open pieces in two basic ways: composition, where we identify
the outputs of one open Markov process with the inputs of another, and
tensoring, where we set two open Markov processes side by side. In
previous work, Fong, Pollard and the first author showed that these
constructions make open Markov processes into the morphisms of a
symmetric monoidal category. Here we go further by constructing a
symmetric monoidal *double* category where the 2-morphisms
include ways of coarse-graining open Markov processes. We also extend
the already known `black-boxing' functor from the category of open
Markov processes to our double category. Black-boxing sends any open
Markov process to the linear relation between input and output data
that holds in steady states, including nonequilibrium steady states
where there is a nonzero flow of probability through the process. To
extend black-boxing to a functor between double categories, we need to
prove that black-boxing is compatible with coarse-graining.

Keywords:
double category, cospan, Markov process, coarse-graining, network,
black box

2010 MSC:
16B50, 18D05, 18D10, 60J27

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 39, pp 1223-1268.

Published 2018-12-13.

http://www.tac.mta.ca/tac/volumes/33/39/33-39.pdf

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