Does every continuous time minimal Markov chain have the Feller property? Consider a Q-matrix on a countable state space. (A Q-matrix is a matrix whose rows sum up to $0$, with nonpositive finite diagonal entries and nonnegative off-diagonal entries.) As explained for example in the book of Norris on Markov chains, every Q-matrix (without any further assumptions) defines a transition function (given by the minimal nonnegative solution of the corresponding backward equation). Is the the semigroup associated to this minimal transition function always Feller? (definition of Feller: functions vanishing at infinity are mapped into functions vanishing at infinity). I know that in general, not every continuous time Markov chain is Feller, but I guess that for these minimal chains it is always true without further assumptions on $Q$. Though I didn't find a statement like this in the books I consulted. Does anybody have a reference/proof/counterexample?

Not every continuous time minimal Markov chain has the Feller property. See a counterexample.