A note on the strong Feller property of diffusion processes

Abstract

In this note we prove the strong Feller property of a strong Markov quasi diffusion process corresponding to an elliptic operator with merely bounded measurable coefficients. We also prove H\"older continuity of harmonic functions associated with the quasi diffusion process and Harnack inequality. As an application, we show that for such diffusion processes the probabilistic definition of a regular boundary point coincides with the 'analytic' one. The parabolic counterparts of these results are presented as well. The proofs are adaptations of arguments from \cite{KrS_79} and \cite{Kr_18}.