(abbreviated as SDE): $$\begin{aligned}
\dif \dot X_t=b_t(X_t,\dot X_t)\dif t+\sigma_t(X_t,\dot X_t)\dif W_t,\ \ (X_0,\dot X_0)=(x,\v)\in\mR^{2d},\end{aligned}$$ where $b_t(x,\v):\mR_+\times\mR^{2d}\to\mR^d$ and $\sigma_t(x,\v):\mR_+\times\mR^{2d}\to\mR^d\otimes\mR^d$ are two Borel measurable functions, $\dot X_t$ denotes the first order derivative of $X_t$ with respect to $t$, and $W_t$ is a $d$-dimensional standard Brownian motion on some probability space $(\Omega,\sF,\mP)$. When $\sigma=0$, the above equation is the classical Newtonian mechanic equation, which describes the motion of a particle. When $\sigma\not=0$, it means that the motion is perturbed by some random external force. More backgrounds about the above stochastic Hamiltonian system are referred to [@So; @Ta], etc.

It is noticed that if we let $Z_t:=(X_t,\dot X_t)$, then $Z_t$ solves the following [*one order*]{} (degenerate) SDE: $$\begin{aligned}
\label{SDE0}
\dif Z_t=(\dot X_t,b_t(Z_t))\dif t+(0,\sigma_t(Z_t)\dif W_t),\ \ Z_0=z=(x,\v)\in\mR^{2d},\end{aligned}$$ and whose time-dependent infinitesimal generator is given by $$\begin{aligned}
\label{LL}
\sL^{a,b}_t f(x,\v):=\tr(a_t\cdot\nabla^2_\v f)(x,\v)+(\v\cdot\nabla_x f)(x,\v)+(b_t\cdot\nabla_\v f)(x,\v).\end{aligned}$$ Here $a_t(x,\v):=\tfrac{1}{2}(\sigma_t\sigma^*_t)(x,\v)$, $\nabla^2_\v f(x,\v)$ stands for the Hessian matrix, the asterisk and tr($\cdot$) denote the transpose and the trace of a matrix respectively. Moreover, let $\mu_t$ be the probability distributional measure of $Z_t$ in $\mR^{2d}$. By Itô’s formula, one knows that $\mu_t$ solves the following Fokker-Planck equation in the distributional sense: $$\begin{aligned}
\label{FP}
\p_t\mu_t=(\sL^{a,b}_t)^*\mu_t,\ \ \mu_0=\delta_z,\end{aligned}$$ where $\delta_z$ is the Dirac measure at 