A monotone convergence theorem for strong Feller semigroups

Abstract For an increasing sequence $$(T_n)$$ ( T n ) of one-parameter semigroups sub Markovian kernel operators over a Polish space, we study the limit semigroup and prove sufficient conditions for it to be strongly Feller. In particular, show that strong Feller property carries from approximating if resolvent latter maps $$\mathbb {1}$$ 1 continuous function. This is instrumental in elliptic on {R}^d$$ R d with unbounded coefficients: our abstract result enables us assign such operator under very mild regularity assumptions coefficients. We also provide counterexamples demonstrate main are close optimal.