coordinate process over $\Omega$. Hence, in order to prove Theorem \[Main2\], it suffices to show the well-posedness of martingale problem for $\sL^{a,b}_t$ in the sense of Stroock and Varadhan [@St-Va]. This will be achieved by proving some Krylov’s type estimate (see Theorem \[Th41\] below), which is also a key tool for proving Theorem \[Main\]. It is remarked that in [@Ro-Zh], we have already used this technique to show the uniqueness of measure-valued solutions and $L^p$-solutions to possibly degenerate second order Fokker-Planck equations under some weak conditions on the coefficients (but not the case of Theorem \[Main2\]).

This paper is organized as follows: in Section 2, we introduce some anisotropic fractional Bessel potential spaces, and prepare some useful estimates for later use. In Section 3, we show the $L^p$-maximal regularity estimate for kinetic Fokker-Planck equations. In Section 4, we study the martingale problem associated with $(\sigma,b)$ under the same assumptions as in Theorem \[Main2\] by showing the basic Krylov’s type estimate. In particular, we first prove Theorem \[Main2\]. In Section 5, we then prove Theorem \[Main\] by using Zvonkin’s transformation and Krylov’s estimate obtained in the previous section. In Appendix, a stochastic Gronwall’s type lemma used in Section 5 is given.

