Stochastic Hamiltonian flows with singular coefficients

Abstract

In this paper, we study the following stochastic Hamiltonian system in ℝ^2d (a second order stochastic differential equation):

$$d{\dot X_t} = b({X_t},{\dot X_t})dt + \sigma ({X_t},{\dot X_t})d{W_t},({X_0},{\dot X_0}) = (x,v) \in \mathbb{R}^{2d},$$

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 ^2/3,0 _p and ∇σ ∈ L^p for some p > 2(2d+1), where H ^{α, β} _p 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).