Stochastic Hamiltonian flows with singular coefficients Dedicated to the 60th Birthday of Professor Michael R¨ockner

In this paper, we study the following stochastic Hamiltonian system in R^(2d)(a second order stochastic differential equation):dX_t = b(X_t,X_t)dt + σ(X_t,X_t)dW_t,(X_0,X_0) =(x, v) ∈ R^(2d),where b(x, v) : R^(2d)→ R^d and σ(x, v) : R^(2d)→ R^d ? R^d are two Borel measurable functions. We show that if σ is bounded and uniformly non-degenerate, and b ∈ H_p^(2/3,0) and ?σ∈ L^p for some p > 2(2 d + 1), where H_p^(α,β)is the Bessel potential space with differentiability indices α in x and β in v, then the above stochastic equation admits a unique strong solution so that(x, v) → Z_t(x, v) :=(Xt,Xt)(x, v) forms a stochastic homeomorphism flow,and(x, v) → Z_t(x, v) is weakly differentiable with ess.sup_(x,v)E(sup_(t∈[0,T])|?Z_t(x, v)|~q) < ∞ for all q ≥ 1 and T≥ 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(2008) and Trevisan(2016).