  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. We
establish its Feller properties and examine how to linearize the associated
sublinear Markovian semigroup. In particular, we observe a novel smoothing
effect of sublinear semigroups in frameworks which carry enough randomness.
Furthermore, we link the value function corresponding to the semigroup to a
nonlinear Kolmogorov equation. This provides a connection to the so-called
Nisio semigroup.
