  In this paper we establish the optimal regularity estimates for the Cauchy
problem of stochastic kinetic equations with random coefficients in anisotropic
Besov spaces. As applications, we study the nonlinear filtering problem for a
degenerate diffusion process, and obtain the existence and regularity of
conditional probability densities under few assumptions. Moreover, we also show
the well-posedness for a class of super-linear growth stochastic kinetic
equations driven by velocity-time white noises, as well as a kinetic version of
Parabolic Anderson Model with measure as initial values.
