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In this note we connect the notion of solutions of a martingale problem tothe notion of a strongly continuous and locally equi-continuous semigroup onthe space of bounded continuous functions equipped with the strict topology.This extends the classical connection of semigroups to Markov processes thatwas used successfully in the context of compact spaces to the context of Polishspaces.  In addition, we consider the context of locally compact spaces and show howthe transition semigroup on the space of functions vanishing at infinity can beextended to the space of bounded continuous functions.

Strict continuity of the transition semigroup for the solution of a well-posed martingale problem