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abstract: 'In this paper we study the following stochastic Hamiltonian system in $\mR^{2d}$ (a second order stochastic differential equation), $$\dif \dot X_t=b(X_t,\dot X_t)\dif t+\sigma(X_t,\dot X_t)\dif W_t,\ \ (X_0,\dot X_0)=(x,\v)\in\mR^{2d},$$ where $b(x,\v):\mR^{2d}\to\mR^d$ and $\sigma(x,\v):\mR^{2d}\to\mR^d\otimes\mR^d$ 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.$\sup_{x,\v}\mE\left(\sup_{t\in[0,T]}|\nabla Z_t(x,\v)|^q\right)<\infty$ 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 [@Fi] and Trevisan [@Tre].'
address: |
     School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, P.R.China\
    Email: XichengZhang@gmail.com 
author:
- |
    XICHENG ZHANG\
    \
    [*Dedicated to the 60th birthday of Professor Michael Röckner*]{}
title: Stochastic Hamiltonian flows with singular coefficients
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Introduction
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Consider the following [*second order*]{} time dependent stochastic differential equation 