Stochastic Hamiltonian flows with singular coefficients

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, (X₀, X˙ ₀) = (x, v) ∈ R^{2 d}, 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(2d+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) := (X_t, Ẋ_t)(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).