Science China Mathematics

, Volume 61, Issue 8, pp 1353–1384 | Cite as

Stochastic Hamiltonian flows with singular coefficients

  • Xicheng Zhang


In this paper, we study the following stochastic Hamiltonian system in ℝ2d (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) : ℝ2d → ℝd and σ(x; v): ℝ2d → ℝd ⊗ ℝd are two Borel measurable functions. We show that if σ is bounded and uniformly non-degenerate, and bH p 2/3,0 and ∇σLp 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) ↦ Zt(x, v) := (Xt, t)(x, v) forms a stochastic homeomorphism flow, and (x, v) ↦ Zt(x, v) is weakly differentiable with ess.supx, v E(supt∈[0, T] |∇Zt(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).


stochastic Hamiltonian system weak differentiability Krylov’s estimate Zvonkin’s transformation kinetic Fokker-Planck operator 




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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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