arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1606.04360math.PRmath.AP
收藏

具有奇异系数的随机Hamilton流

Stochastic Hamiltonian flows with singular coefficients

Xicheng Zhang

备注：
40pages
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

本文研究了二阶随机微分方程${\mathbb R}^{2d}$中的随机Hamilton系统$$ d \dot X_t=b(X_t,\dot X_t)d t+σ(X_t,\dot X_t)d W_t,\ \ (X_0,\dot X_0)=(x,v)\in{\mathbb R}^{2d}, $$，其中$b(x,v):{\mathbb R}^{2d}\to{\mathbb R}^d$和$σ(x,v):{\mathbb R}^{2d}\to{\mathbb R}^d\otimes{\mathbb R}^d$是两个Borel可测函数.证明了若$σ$是有界的一致非退化的，且$b\in H^{2/3,0}_p$和$\nablaσ\in L^p$对于某个$p>2(2d+1)$，其中$H^{α,β}_p$是Bessel位势空间，其可微性指数$α$在$x$中，$β$在$v$中，则上述随机方程存在唯一强解，使得$(x,v)\mapsto Z_t(x,v):=(X_t,\dot X_t)(x,v)$形成随机同胚流，对所有的$q\geq 1$和$T\geq 0$，$(x,v)\mapsto Z_t(x,v)$与ess. $\sup_{x,v}E\left(\sup_{t\in[0,T]}|\nabla Z_t(x,v)|^q\right)<\infty$弱可微.此外，通过证明相应鞅问题的适定性，利用Figalli \cite{Fi}和Trevisan \cite{Tre}建立的叠加原理，我们还证明了具有粗糙系数的动力学Fokker-Planck方程概率测度解的唯一性.

英文摘要：

In this paper we study the following stochastic Hamiltonian system in ${\mathbb R}^{2d}$ (a second order stochastic differential equation), $$ d \dot X_t=b(X_t,\dot X_t)d t+σ(X_t,\dot X_t)d W_t,\ \ (X_0,\dot X_0)=(x,v)\in{\mathbb R}^{2d}, $$ where $b(x,v):{\mathbb R}^{2d}\to{\mathbb R}^d$ and $σ(x,v):{\mathbb R}^{2d}\to{\mathbb R}^d\otimes{\mathbb R}^d$ are two Borel measurable functions. We show that if $σ$ is bounded and uniformly non-degenerate, and $b\in H^{2/3,0}_p$ and $\nablaσ\in L^p$ 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 $(x,v)\mapsto Z_t(x,v):=(X_t,\dot X_t)(x,v)$ 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 $q\geq 1$ and $T\geq 0$. 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}.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑