In this talk we will study kinetic equations with multiple scales

and random uncertainties from initial data and/or collision kernel. Here the multiple scales, characterized by the Knudsen number, will lead the kinetic equations to hydrodynamic (Euler, incompressible Navier-Stokes or diffusion) equations as the Knudsen number goes to zero. Asymptotic-preserving schemes, which minic the asymptotic transitions from the microscopic to the macroscopic scales at the discrete level, have been shown to be effective to deal with multiscale problems in the deterministic setting.

We first extend the prodigm of asymptotic-preserving schemes to the random kinetic equations, and show how it can be constructed in the setting of the stochastic Galerkin approximations. We then extend the hypocoercivity theory, developed for deterministic kinetic equations, to the random case, and establish in the random space regularity, long-time sensitivity analysis, and uniform (in Knudsen number) spectral convergence of the stochastic Galerkin methods, for general lienar and nonlinear random kinetic equations in various asymptotic--including the duffusion, incompressible Navier-Stokes, high-field, and acoustic-- regimes.