具有无界Hölder连续漂移的随机微分方程的自同态流

Drift reduction method for SDEs driven by inhomogeneous singular L{é}vy noise

We study SDE $$ d X_t = b(X_t) \, dt + A(X_{t-}) \, d Z_t, \quad X_{0} = x \in \mathbb{R}^d, \quad t \geq 0 $$ where $Z=(Z^1, \dots, Z^d)^T$, with $Z^i, i=1,\dots, d$ being independent one-dimensional symmetric jump Lévy processes, not necessarily identically distributed. In particular, we cover the case when each $Z^i$ is one-dimensional symmetric $α_i$-stable process ($α_i \in (0,2)$ and they are not necessarily equal). Under certain assumptions on $b$, $A$ and $Z$ we show that the weak solution to the SDE is uniquely defined and Markov, we provide a representation of the transition probabil…

Heat kernel of supercritical SDEs with unbounded drifts

Let $α\in(0,2)$ and $d\in{\mathbb N}$. Consider the following SDE in ${\mathbb R}^d$:$${\rm d}X_t=b(t,X_t){\rm d} t+a(t,X_{t-}){\rm d} L^{(α)}_t,\ \ X_0=x,$$where $L^{(α)}$ is a $d$-dimensional rotationally invariant $α$-stable process, $b:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ and $a:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ are H{ö}lder continuous functions in space, with respective order $β,γ\in (0,1)$ such that $(β\wedge γ)+α>1$, uniformly in $t$. Here $b$ may be unbounded.When $a$ is bounded and uniformly elliptic, we show that the unique solution $X…

具有奇异漂移的分数布朗运动驱动的随机微分方程的强解

Strong existence for SDEs with singular drifts driven by fractional Brownian motion

本文研究了由分数布朗运动驱动的具有奇异漂移的随机微分方程的强解存在性，扩展了经典结果并证明了 Sobolev 偏微分同胚的随机流存在性。

完全次临界机制下分数阶SDE的解理论

Solution theory of fractional SDEs in complete subcritical regimes

本文研究了由分数布朗运动驱动的SDEs，其漂移系数在尺度意义上可任意接近临界值，并提出包含强存在性、逐路径唯一性、解流存在性、Malliavin可微性及$ρ$-不规则性的完整解理论，进而可处理McKean-Vlasov等方程。

具有奇异系数的随机微分方程的Sobolev可微随机流：在输运方程中的应用

Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation

本文证明了形式为\[dX_t=b(t,X_t)\,dt+dB_t,\qquad s,t\in\mathbb{R},X_s=x\in\mathbb{R}^d.\]的随机微分方程的Sobolev同态\[\mathbb{R}^d\ni x\quad\longmapsto\quadϕ_{s,t}(x)\in \mathbb{R}^d,\qquad s,t\in\mathbb{R}\]的随机流的存在性。上述随机流由有界可测漂移系数$b:\mathbb{R}\times\mathbb{R}^d\rightarrow\mathbb{R}^d$和$d$维布朗运动$B$驱动。更具体地说，我们证明了随机流$ϕ_{s,t}(\cdot)$的生活在空间$L^2(Ω;W^{1,p}(\mathbb{R}^d,w))$的所有$s,t$和所有$p\in (1,\infty)$，其中$W^{1,p}(\mathbb{R}^d,w)$表示加权Sobolev空间的重量$w$拥有$p$次矩关于Lebesgue测量$\mathbb {R}^d$。从随机（和确定性）动力系统的观点来看，这是一个惊人的结果，因为在这些动力系统中占主导地位的“文化”是流“继承”其空间规律性的驱动矢量场。随机流的空间正则性给出了（Stratonovich）随机迁移方程\[\cases{\displaystyle d_tu(t,x)+\bigl(b…

带Hölder漂移的随机流及带噪声的输运/连续性方程

Stochastic flows for Hölder drifts and transport/continuity equations with noise

该研究通过Zvonkin型变换证明了带Hölder漂移SDE的微分同胚随机流存在性，建立相关抛物PDE的适定性与最优正则性，进而得到带输运噪声的随机输运与连续性方程的适定性，弥补确定性情形下解可能不唯一的问题。