  In this paper we study the following stochastic Hamiltonian system in
${\mathbb R}^{2d}$ (a second order stochastic differential equation), $$ d \dot
X_t=b(X_t,\dot X_t)d t+\sigma(X_t,\dot X_t)d W_t,\ \ (X_0,\dot
X_0)=(x,v)\in{\mathbb R}^{2d}, $$ where $b(x,v):{\mathbb R}^{2d}\to{\mathbb
R}^d$ and $\sigma(x,v):{\mathbb R}^{2d}\to{\mathbb R}^d\otimes{\mathbb R}^d$
are two Borel measurable functions. We show that if $\sigma$ is bounded and
uniformly non-degenerate, and $b\in H^{2/3,0}_p$ and $\nabla\sigma\in L^p$ for
some $p>2(2d+1)$, where $H^{\alpha,\beta}_p$ is the Bessel potential space with
differentiability indices $\alpha$ in $x$ and $\beta$ in $v$, then the above
stochastic equation admits a unique strong solution so that $(x,v)\mapsto
Z_t(x,v):=(X_t,\dot X_t)(x,v)$ forms a stochastic homeomorphism flow, and
$(x,v)\mapsto Z_t(x,v)$ is weakly differentiable with
ess.$\sup_{x,v}E\left(\sup_{t\in[0,T]}|\nabla Z_t(x,v)|^q\right)<\infty$ for
all $q\geq 1$ and $T\geq 0$. Moreover, we also show the uniqueness of
probability measure-valued solutions for kinetic Fokker-Planck equations with
rough coefficients by showing the well-posedness of the associated martingale
problem and using the superposition principle established by Figalli \cite{Fi}
and Trevisan \cite{Tre}.
