Is the infinitesimal generator for Lie groups related to the infinitesimal generator of a Markov semigroup? Or are they totally different concepts?
https://en.wikipedia.org/wiki/Lie_group#The_exponential_map https://en.wikipedia.org/wiki/Infinitesimal_generator_(stochastic_processes)
Both seem to be related to the exponential map.
The connection would also explain why so many people, when discussing infinitesimal generators of a Markov process, seem to have such a strong intuition for it, when it seems like such a painfully abstract concept. However, if they are already familiar with the concept as applied to much simpler objects (i.e. matrix groups in Lie theory) that might explain the apparent discrepancy between understanding and presentation.
Nevertheless I have never heard this connection made explicit before, besides both subjects being mentioned on the same disambiguation page on Wikipedia.