Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus

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 solutions to SDEs with singular drifts driven by fractional Brownian motions

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。

具有临界时间依赖漂移项的随机微分方程：强解

SDEs with critical time dependent drifts: strong solutions

本文利用 Wiener-Sobolev 空间中的随机场紧性准则，证明了漂移系数位于临界 Lebesgue 空间的时间非齐次随机微分方程具有唯一强解，并应用于与 Navier-Stokes 方程相关的 Constantin-Iyer 随机系统正则性准则。

A stochastic-Lagrangian particle system for the Navier-Stokes equations

This paper is based on a formulation of the Navier-Stokes equations developed by P. Constantin and the first author (\texttt{arxiv:math.PR/0511067}, to appear), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with $\frac{1}{N}$ times the sum over these $N$ copies. (We remark that our formulation requires one to keep track of $N$ stochastic flows of diffeomorphisms, and not just the motion of $N$ …

McKean-Vlasov SDE 的复合泊松粒子逼近

Compound Poisson particle approximation for McKean-Vlasov SDEs

本文提出基于复合泊松粒子的时空同步离散化格式，证明 McKean-Vlasov SDE 的混沌传播与收敛性，并给出 ODE 最优速率、CLT 及二维 Navier-Stokes 应用。

Generalized stochastic Lagrangian paths for the Navier-Stokes equation

In the note added in proof of the seminal paper [Groups of diffeomorphisms andthe motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163], Ebinand Marsden introduced the so-called correct Laplacian for the Navier-Stokes equationon a compact Riemannian manifold. In the spirit of Brenier's generalized flows forthe Euler equation, we introduce a class of semimartingales on a compact Riemannianmanifold. We prove that these semimartingales are critical points to the correspondingkinetic energy if and only if its drift term solves weakly the Navier-Stokes equationdefined with Ebin-Marsd…

随机保体积微分同胚的负正则性混合

Negative regularity mixing for random volume preserving diffeomorphisms

本文研究随机保体积微分同胚的负正则性指数混合，给出通用判据并应用于随机微分同胚、SDE 及随机不可压缩 Navier-Stokes 平流扩散，证明理想标量湍流存在唯一平稳测度。

麦肯-瓦拉斯方程组的随机吸引子

Random Attractors for McKean-Vlasov SDEs

本文研究了麦肯-瓦拉斯随机微分方程组在可分希尔伯特空间上的随机吸引子存在性，通过转换方程到产品空间并分析协程，建立了通用理论并应用于多种随机偏微分方程。

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。

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

In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms \[\mathbb{R}^d\ni x\quad\longmapsto\quadϕ_{s,t}(x)\in \mathbb{R}^d,\qquad s,t\in\mathbb{R}\] for a stochastic differential equation (SDE) of the form \[dX_t=b(t,X_t)\,dt+dB_t,\qquad s,t\in\mathbb{R},X_s=x\in\mathbb{R}^d.\] The above SDE is driven by a bounded measurable drift coefficient $b:\mathbb{R}\times\mathbb{R}^d\rightarrow\mathbb{R}^d$ and a $d$-dimensional Brownian motion $B$. More specifically, we show that the stochastic flow $ϕ_{s,t}(\cdot)$ of the SDE lives in the space $L^2(Ω;W^{1,p}(\mathbb{…