In this paper we study a family of nonlinear (conditional) expectations that
can be understood as a diffusion with uncertain local characteristics. Here,
the differential characteristics are prescribed by a set-valued function that
depends on time and path in a Markovian way. We establish its Feller properties
and examine how to linearize the associated sublinear Markovian semigroup. In
particular, we observe a novel smoothing effect of nonlinear semigroups in
frameworks which carry enough randomness. Furthermore, we link the value
function corresponding to the semigroup to a nonlinear Kolmogorov equation