for any $z=(x,\v)\in\mR^{2d}$, SDE admits a unique strong solution $Z_t(z)=(X_t,\dot X_t)$ so that $(t,z)\mapsto Z_t(z)$ has a bi-continuous version. Moreover,

1.  There is a null set $N$ such that for all $\omega\notin N$ and for each $t\geq 0$, the map $z\mapsto Z_t(z,\omega)$ is a homeomorphism on $\mR^{2d}$.

2.  For each $t\geq 0$, the map $z\mapsto Z_t(z)$ is weakly differentiable a.s., and for any $q\geq 1$ and $T>0$, $$\begin{aligned}
    \label{YN1}
    \mathrm{ess.}\sup_{z}\mE\left(\sup_{t\in[0,T]}|\nabla Z_t(z)|^q\right)<\infty,\end{aligned}$$ where $\nabla$ denotes the generalized gradient.

3.  Let $\sigma^n$ and $b^n$ be the regularized approximations of $\sigma$ and $b$ (see below for definitions). Let $Z^n$ be the corresponding solution of SDE associated with $(\sigma^n,b^n)$. For any $q\geq 1$ and $T>0$, there exits a constant $C>0$ only depending on $T,K,\kappa_0,d,p,q$ such that $$\mE\left(\sup_{t\in[0,T]}|Z^n_t-Z_t|^q\right)\leq C\left(\|b^n-b\|^q_{\mL^p(T)}+n^{(\frac{2d}{p}-1)q}\right),\ \ n\in\mN.$$

As a corollary, we have the following local well-posedness result by a standard localization argument. Suppose that for any $T, R>0$, there exists a constant $K_{T,R}\geq 1$ such that for all $(t,x,\v)\in[0,T]\times B_R$ and $\xi\in\mR^d$, $$\begin{aligned}
K_{T,R}^{-1}|\xi|\leq |\sigma^*_t(x,\v)\xi|\leq K_{T,R}|\xi|,\label{Con22}\end{aligned}$$ where $B_R:=\{(x,\v): |(x,\v)|\leq R\}$, and for some $p>2(2d+1)$, $$\sup_{t\in[0,T]}\|\nabla(\sigma_t\chi_R)\|^p_{p}+\int_0^T\|(\mI-\Delta_x)^{\frac{1}{3}} (b_s\chi_R)\|_p^p\dif s\leq K_{T,R},$$ where $\chi_R:\mR^{2d}\to[0,1]$ is a smooth function with $\chi_R(z)=1$ for $|z|\leq R$ and $\chi_R(z)=0$ for $|z|>2R$. Then for any fixed $(x,\v)\in\mR^{2d}$, SDE admits a 