Strongly-Feller Processes I. General Properties

In this paper a class of Markov processes on the metric space E is investigated. A process is called strongly-Feller (respectively, strongly-Feller in the narrow sense) if for every bounded measurable function f and for $t > s$,\[ Tₛₜ f(x) = { M}s,x f( x_t ) = ∫ {{ P}(s,x;t,dy)f(y)} \] is continuous in x (respectively, if \[ {Var}|{ P}(s,x;t,Γ ) - { P}(s,y;t,Γ )| → 0 \] as y → x. Continuity is proved for a wide class of functions of the type \[ φ (x) = { M}s,x ξ (ω ). \] For homogeneous processes with continuous trajectories which are strongly-Feller in the narrow sense, the continuity of the functions \[ φ ( x ) = { M}_x f( {xτ _Γ } ), ψ ( x ) = { M}_x g( {∫_0τ _Γ {V( {x_t } )} dt} ) \] on the boundary Γ is investigated. The results are applied to the investigation of various functionals of the process X and to the investigation of the subprocesses of X.