Stochastic Hamiltonian flows with singular coefficients

Abstract

In this paper we study the following stochastic Hamiltonian system in ${\mathbb{R}}^{2d}$ (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 $b(x,v):{\mathbb{R}}^{2d}\to {\mathbb{R}}^d$ and $\sigma (x,v):{\mathbb{R}}^{2d}\to {\mathbb{R}}^d\otimes {\mathbb{R}}^d$ are two Borel measurable functions. We show that if $\sigma$ is bounded and uniformly non-degenerate, and $b\in H_p^{2/3,0}$ and $\nabla \sigma \in L^p$ for some $p>2(2d+1)$, where $H_p^{\alpha ,\beta }$ 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.${\sup }_{x,v}E\left({\sup }_{t\in [0,T]}|\nabla Z_t(x,v){|}^q\right)<\infty$ for all $q\ge 1$ and $T\ge 0$. 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 \cite{Fi} and Trevisan \cite{Tre}.

40pages

References in corpus (3)

• Noise Prevents Singularities in Linear Transport Equations
• $L^p$-maximal hypoelliptic regularity of nonlocal kinetic Fokker-Planck operators
• Martingale problems for some degenerate Kolmogorov equations

Cited by in corpus (4)

• Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli
• Quantitative stability estimates for Fokker-Planck equations
• Estimates for Invariant Probability Measures of Degenerate SPDEs with Singular and Path-Dependent Drifts
• Distribution-Dependent Stochastic Functional Differential Equations