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Jacob Biamonte and I just finished our draft for a blog article about a Noether's theorem for doubly stochastic Markovian systems. We show that in a continuous-time Markovian system where the generator is symmetric (and thus a Dirichlet operator), one regains the usual "quantum version" of the Noether's theorem instead of the more complicated "stochastic version" John and Brendan presented in a recent paper.
You can find the article here:
We feel it's pretty close to finished, all comments are welcome.