  Gaseous flows show a diverse set of behaviors on different characteristic
scales. Given the coarse-grained modeling in theories of fluids, considerable
uncertainties may exist between the flow-field solutions and the real physics.
To study the emergence, propagation and evolution of uncertainties from
molecular to hydrodynamic level poses great opportunities and challenges to
develop both sound theories and reliable multi-scale numerical algorithms. In
this paper, a new stochastic kinetic scheme will be developed that includes
uncertainties via a hybridization of stochastic Galerkin and collocation
methods. Based on the Boltzmann-BGK model equation, a scale-dependent evolving
solution is employed in the scheme to construct governing equations in the
discretized temporal-spatial domain. Therefore typical flow physics can be
recovered with respect to different physical characteristic scales and
numerical resolutions in a self-adaptive manner. We prove that the scheme is
formally asymptotic-preserving in different flow regimes with the inclusion of
random variables, so that it can be used for the study of multi-scale
non-equilibrium gas dynamics under the effect of uncertainties.
  Several numerical experiments are shown to validate the scheme. We make new
physical observations, such as the wave-propagation patterns of uncertainties
from continuum to rarefied regimes. These phenomena will be presented and
analyzed quantitatively. The current method provides a novel tool to quantify
the uncertainties within multi-scale flow evolutions.
