The Feller property on Riemannian manifolds

The asymptotic behavior of the heat kernel of a Riemannian manifold gives rise
to the classical concepts of parabolicity, stochastic completeness (or
conservative property) and Feller property (or $C^{0}$-diffusion property).
Both parabolicity and stochastic completeness have been the subject of a
systematic study which led to discovering not only sharp geometric conditions
for their validity but also an incredible rich family of tools, techniques and
equivalent concepts ranging from maximum principles at infinity, function
theoretic tests (Khas'minskii criterion), comparison techniques etc... The
present paper aims to move a number of steps forward in the development of a
similar apparatus for the Feller property.