Define for $t<s$, $$\begin{aligned}
\label{PP}
P_{t,s}f(x,\v)=\mE f(x+(s-t)\v+X_{t,s}, \v+V_{t,s}),\end{aligned}$$ where $$(X_{t,s},V_{t,s})=\left(\int^s_tV_{t,r}\dif r, \int^s_t\sigma_r\dif W_r\right).$$

We need the following basic $L^p$-regularity estimates related to $P_{t,s}$, which plays a basic role in the next section. \[Th25\] Let $T>0$. Suppose that for some $K>0$ and all $t\in[0,T]$, $$K^{-1}|\xi|\leq |\sigma^*_t\xi|\leq K|\xi|,\ \ \xi\in\mR^d.$$

(i) For any $\alpha,\beta\geq 0$ and $p>1$, there exists a positive constant $C=C(K,T,p,d,\alpha,\beta)$ such that for all $f\in L^p(\mR^{2d})$ and $0\leq t<s\leq T$, $$\begin{aligned}
    \label{Gr}
    \begin{split}
    \|P_{t,s}f\|_{\alpha,0;p}\leq C(s-t)^{-\frac{3\alpha}{2}}\|f\|_p,\\
    \|P_{t,s}f\|_{0,\beta;p}\leq C(s-t)^{-\frac{\beta}{2}}\|f\|_p.
    \end{split}\end{aligned}$$

(ii) For any $p>1$, there exists a positive constant $C_p=C_p(K,d)$ such that for all $\lambda\geq 0$ and $f\in\mL^p(T)=L^p([0,T]\times\mR^{2d})$, $$\begin{aligned}
    \label{Pri22}
    \|\nabla^2_\v u^\lambda\|_{\mL^P(T)}+\|\Delta^{\frac{1}{3}}_x u^\lambda\|_{\mL^P(T)}\leq C_p\|f\|_{\mL^P(T)},\end{aligned}$$ where $u^\lambda_t(x,\v):=\int^T_t\e^{\lambda(t-s)}P_{t,s}f_s(x,\v)\dif s$ satisfies $$\p_t u^\lambda+\sL^{a,0}_tu^\lambda-\lambda u^\lambda+f=0$$ in the distributional sense.

\(i) It follows by the following gradient estimate and the interpolation theorem (see [@Wa-Zh Theorem 2.10]), $$\|\nabla^k_x\nabla^m_\v P_{t,s}f\|_p\leq C(s-t)^{-\frac{3k+m}{2}}\|f\|_p,\ k,m\in\mN_0.$$ (ii) It is a consequence of [@Br-Cu-La-Pr] and [@Bo Theorem 2.1] (see also [@Ch-Zh Theorem 3.3]).

Notice that in the references [@Wa-Zh] and [@Ch-Zh], the positions of $t$ and $s$ are exchanged.

Maximal $L^p$-solutions of kinetic Fokker-Planck equations
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Throughout this section, we fix $T>0$. Let $p\in(1,\infty)$ and $\alpha,\beta\geq 0$. For $t\in[0,T]$, we introduce the following Banach spaces with natural norms: $$\mL^p(t,T):=L^p([t,T];L^p(\mR^{2d})),\ \ \mH^{\alpha,\beta}_p(t,T):=L^p([t,T];H^{\alpha,\beta}_p(\mR^{2d})).$$ For simplicity 