Wei-Xi Li, Juan Zeng
We consider a Fokker-Planck operator with electric potential and electromagnetic fields. We establish the sharp weighted and subelliptic estimates, involving the control of the derivatives of electric potential and electromagnetic fields. Our proof relies on a localization argument as well as a careful calculation on commutators.
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In this paper, we study the Gevrey regularity of weak solution for a class of linear and quasilinear Fokker-Planck equations.
Trend to equilibrium and hypoelliptic regularity for the relativistic Fokker-Planck equation
针对带外部约束势的相对论空间非齐次Fokker-Planck方程，通过构造李雅普诺夫泛函，证明解在加权L²和索伯列夫空间中指数衰减至平衡且收敛速率显式，同时证明半群具亚椭圆正则性并获显式正则化速率。
We prove sharp Harnack inequalities for a family of Kolmogorov-Fokker-Planck type hypoelliptic diffusions.
In this paper we study the Fokker-Planck operator with potential V(x), and analyze some kind of conditions imposed on the potential to ensure the validity of global hypoelliptic estimates. As a consequence, we obtain the compactness of resolvent of the Fokker-Planck operator if either the Witten Laplacian on 0-forms has a compact resolvent or some additional assumption on the behavior of the potential at infinity is fulfilled. This work improves the previous results of Hérau-Nier and Helffer-Nier, by obtaining a better global hypoelliptic estimate under weaker assumptions on the potential.
Exponential stability and hypoelliptic regularization for the kinetic Fokker-Planck equation with confining potential
本文通过改进的熵方法，利用非常数权重矩阵的 Lyapunov 泛函，证明了带非二次约束势的动理学 Fokker-Planck 方程在加权 $H^1$-范数下的指数收敛性，并给出缺陷二次势情形的精确衰减估计及新的亚椭圆正则化结果。
In this work, Holder continuity is obtained for solutions to the nonlocal kinetic Fokker-Planck Equation, and to a family of related equations with general integro-differential operators. These equations can be seen as a generalization of the Fokker-Planck Equation, or as a linearization of non-cutoff Boltzmann. Difficulties arise because our equations are hypoelliptic, so we utilize the theory of averaging lemmas. Regularity is obtained using De Giorgi's method, so it does not depend on the regularity of initial conditions or coefficients. This work assumes stronger constraints on the nonloca…
The purpose of this paper is to extend the hypocoercivity results for the kinetic Fokker-Planck equation in $H^1$ space in Villani's memoir \cite{Villani} to higher order Sobolev spaces. As in the $L^2$ and $H^1$ setting, there is lack of coercivity in $H^k$ for the associated operator. To remedy this issue, we shall modify the usual $H^k$ norm with certain well-chosen mixed terms and with suitable coefficients which are constructed by induction on $k$. In parallel, a similar strategy but with coefficients depending on time (c.f. \cite{Herau}), usually referred as Hérau's method, can be employ…
We give an index formula for the class of all *-maximally hypoelliptic differential operators on any closed manifold with vector bundle coefficients, generalising previous index formulas by Atiyah-Singer and van Erp. Using this formula, we give new explicit index computations for Hormander's sum of squares operators of arbitrary rank.
Sharp boundary regularity properties for hypoelliptic kinetic equations
该研究针对动理学Fokker-Planck方程，在给定流入边界条件下建立了精确的边界次椭圆正则性结果，刻画了不同系数条件下解在各类边界的性态并得到相关估计。
We extend the classical regularity theorem of elliptic operators to maximally hypoelliptic differential operators. More precisely, given vector fields $X_1,\ldots,X_m$ on a smooth manifold which satisfy Hörmander's bracket generating condition, we define a principal symbol for \textit{any} linear differential operator. Our symbol takes into account the vector fields $X_i$ and their commutators. We show that for an arbitrary differential operator, its principal symbol is invertible if and only if the operator is maximally hypoelliptic. This answers affirmatively a conjecture due to Helffer and …
This article is concerned with maximal accretive realizations of geometric Kramers-Fokker-Planck operators on manifolds with boundaries. A general class of boundary conditions is introduced which ensures the maximal accretivity and some global subelliptic estimates. Those estimates imply nice spectral properties as well as exponential decay properties for the associated semigroup. Admissible boundary conditions cover a wide range of applications for the usual scalar Kramer-Fokker-Planck equation or Bismut's hypoelliptic Laplacian.
For $p\in(1,\infty)$, let $u(t,x,v)$ and $f(t,x,v)$ be in $L^p(\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d)$ and satisfy the following nonlocal kinetic Fokker-Plank equation on $\mathbb{R}^{1+2d}$ in the weak sense: $$ \partial_t u+v\cdot\nabla_x u=Δ^{α/{2}}_v u+f, $$ where $α\in(0,2)$ and $Δ^{α/{2}}_v$ is the usual fractional Laplacian applied to $v$-variable. We show that there is a constant $C=C(p,α,d)>0$ such that for any $f(t, x, v)\in L^p(\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d)=L^p(\mathbb{R}^{1+2d})$, $$ \|Δ_x^{α/{(2(1+α))}}u\|_p+\|Δ_v^{α/{2}}u\|_p\leq C\|f\|_p, $$ wh…
We study the initial-boundary value problem for the Fokker-Planck equation in an interval with absorbing boundary conditions. We develop a theory of well-posedness of classical solutions for the problem. We also prove that the resulting solutions decay exponentially for long times. To prove these results we obtain several crucial estimates, which include hypoellipticity away from the singular set for the Fokker-Planck equation with absorbing boundary conditions, as well as the Holder continuity of the solutions up to the singular set.
Weak and renormalized solutions to a hypoelliptic Mean Field Games system
本文证明了退化次椭圆平均场博弈系统弱解与重整化解的存在唯一性，利用De-Giorgi论证和动力学正则性，覆盖Lipschitz及二次增长Hamilton量。
Gehring's Lemma for kinetic Fokker-Planck equations
该研究将Gehring引理推广到动理学次椭圆尺度并实现局部化，给出简洁定量证明，并用其证明带粗糙系数的线性Fokker-Planck方程弱解速度梯度的可积性严格高于平方可积，提升了原有估计。
Anisotropic Maximal $L^p$-regularity Estimates for a Hypoelliptic Operator
本文研究了双曲算子的各向异性最大 $L^p$-正则性估计，证明了解的存在性并给出了相关算子的弱估计。
Long-time contractivity estimates for kinetic Kolmogorov-Fokker-Planck equations
本文针对动力学 Kolmogorov-Fokker-Planck 等亚椭圆扩散方程，提出了一种基于伴随问题振荡估计的初等自包含方法，通过加倍变量和短时间梯度估计，证明了长时间收缩性及指数收敛速率。
This article is concerned with the Schauder estimate for linear kinetic Fokker-Planck equations with Hölder continuous coefficients. This equation has an hypoelliptic structure. As an application of this Schauder estimate, we prove the global well-posedness of a toy nonlinear model in kinetic theory. This nonlinear model consists in a non-linear kinetic Fokker-Planck equation whose steady states are Maxwellian and whose diffusion in the velocity variable is proportional to the mass of the solution.
Unconditional alignment of solutions to the Fokker-Planck-Navier-Stokes system with locally averaged Brinkman force
研究悬浮于粘性流体中粒子的耦合FPNS系统，通过新的熵方法与hypocoercivity框架证明了周期域上所有弱解的粒子与流体速度无条件对齐，并建立了全局弱解与二维强解的适定性理论。
On a spatially inhomogeneous nonlinear Fokker-Planck equation: Cauchy problem and diffusion asymptotics
该研究针对空间非均匀非线性Fokker-Planck方程的动理学模型，证明当初值低于麦克斯韦分布时的全局适定性与瞬时光滑性，建立低场极限下与快扩散流的关联并推导全局定量扩散渐近行为。
We study the kinetic Fokker-Planck equation perturbed by a stochastic Vlasov force term. When the noise intensity is not too large, we solve the Cauchy Problem in a class of well-localized (in velocity) functions. We also show that, when the noise intensity is sufficiently small, the system with prescribed mass admits a unique invariant measure which is exponentially mixing. The proof uses hypocoercive decay estimates and hypoelliptic gains of regularity. At last we also exhibit an explicit example showing that some restriction on the noise intensity is indeed required.
In this paper, we construct unique, local-in-time strong solutions to the Vlasov-Poisson (VP) and Vlasov-Poisson-Fokker-Planck (VPFP) systems subjected to external, spatially regular, white-in-time electromagnetic fields in $\mathbb T^d \times \mathbb R^d$. Initial conditions are taken $H^σ$ with $σ> d/2 + 1$ (in addition to polynomial velocity weights). We additionally show that solutions to the VPFP are instantly $C^\infty_{x,v}$ due to hypoelliptic regularization if the external force fields are smooth. The external forcing arises in the kinetic equation as a stochastic transport in velocit…
We consider hypoelliptic equations of kinetic Fokker-Planck type, also known as Kolmogorov or ultraparabolic equations, with rough coefficients in the drift-diffusion operator. We give novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities. This implies H{ö}lder continuity with quantitative estimates. The paper is self-contained.
Velocity averaging and Hölder regularity for kinetic Fokker-Planck equations with general transport operators and rough coefficients
本文研究具有一般传输算子和粗糙系数的动力学 Fokker-Planck 方程弱解的局部有界性和 Hölder 连续性，通过结合椭圆 De Giorgi-Nash-Moser 理论与速度平均引理的方法，利用扩散与传输的混合效应及亚椭圆结构证明了该结果。
We study the kinetic Fokker-Planck equation on the whole space with a confining potential. We show quantitative rates of exponential convergence to equilibrium in a well chosen Wasserstein-1 distance. We use the Wasserstein-1 version of Harris's theorem introduced by Hairer and Mattingly. We make use of similarities between hypocoercivity and hypoellipticity in order to use Malliavin calculus to see hypocoercivity for this equation on the level of the SDE.
We study accretive quadratic operators with zero singular spaces. These degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in the Schwartz space for any positive time. In this work, we study the short-time asymptotics of the regularizing effect induced by these semigroups. We show that these short-time asymptotics of the regularizing effect depend on the directions of the phase space, and that this dependence can be nicely understood through the structure of the singular space. As a byproduct of these result…
On a Vlasov-Fokker-Planck equation for stored electron beams
本文研究了描述同步加速器储存环中电子束纵向动力学的自洽 Vlasov-Fokker-Planck 方程，证明了全局经典解的存在唯一性、稳态的存在性，并在小束流电流下证明了稳态的局部渐近非线性稳定性。
Global subelliptic estimates for geometric Kramers-Fokker-Planck operators on closed manifolds
本文针对闭流形上的几何Kramers-Fokker-Planck算子，采用动量二进划分、位置局部分析的新方法，证明了高低摩擦两种渐近情形下带常数控制的极大次椭圆估计，引入双指数Sobolev尺度，且方法更易推广到带边流形。
We extend the De Giorgi--Nash--Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the H{ö}lder regularity and establish a Harnack inequality for solutions to a general linear equation of Fokker-Planck type whose coefficients are merely measurable and essentially bounded, i.e. assuming no regularity on the coefficients in order to later derive results for non-linear problems. This general equation has the formal structure of the hypoelliptic equations "of type II" , sometimes also called ultraparabolic…
Fokker-Planck Analysis and Invariant Laws for a Continuous-Time Stochastic Model of Adam-Type Dynamics
本文提出连续时间模型研究自适应随机优化的长期动态，聚焦于偏置校正Adam型方法，分析Fokker-Planck方程并证明不变测度的存在唯一性。
We investigate the existence of steady states and exponential decay for hypocoercive Fokker--Planck equations on the whole space with drift terms that are linear in the position variable. For this class of equations, we first establish that hypoellipticity of its generator and confinement of the system is equivalent to the existence of a unique normalised steady state. These two conditions also imply hypocoercivity, i.e. exponential convergence of the solution to equilibrium. Since the standard entropy method does not apply to degenerate parabolic equations, we develop a new modified entropy m…
Well-posedness and long-time behavior for self-consistent Vlasov-Fokker-Planck equations with general potentials
本文研究了具外部约束势和非线性自洽相互作用的 Vlasov-Fokker-Planck 方程的适定性及长时间行为，通过引入新相互作用核条件，结合 hypocoercivity 理论设计 Lyapunov 泛函，证明了在低正则性下解的指数衰减及稳态稳定性。
We consider the heat equation associated with a class of hypoelliptic operators of Kolmogorov-Fokker-Planck type in dimension two. We explicitly compute the first meaningful coefficient of the small time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants of the optimal control problem associated with the diffusion. This gives a first example of geometric interpretation of the small-time heat kernel asymptotics of non-homogeneous Hörmander operators which are not associated with a sub-Riemannian structure, i.e., whose second-order …
We study the spatially inhomogeneous Landau equations with hard potential in the perturbation setting, and establish the analytic smoothing effect in both spatial and velocity variables for a class of low-regularity weak solutions. This shows the Landau equations behave essentially as the hypoelliptic Fokker-Planck operators. The spatial analyticity relies on a new time-average operator, and the proof is based on a straightforward energy estimate with a careful estimate on the derivatives with respect to the new time-average operator.
Well-posedness of the Kolmogorov-Fokker-Planck equation on bounded domains
研究有界域上具有球面速度的平稳Kolmogorov方程的适定性，包括流入边界和镜面反射条件，并引入通过输运算子定义迹的超椭圆空间，证明带迹的Poincaré型不等式，部分解决最优权重的迹问题。
Large-time behavior of solutions to Vlasov-Poisson-Fokker-Planck equations: from evanescent collisions to diffusive limit
本文研究了Vlasov-Poisson-Fokker-Planck方程在不同碰撞和非线性 regime下的长时间行为，通过非线性椭圆方程和超协调估计分析解的收敛性和衰减特性。
We consider degenerate Kolmogorov-Fokker-Planck operators $$ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u,\qquad (x,t)\in\mathbb{R}^{N+1},N\geq q\geq1 $$ such that the corresponding model operator having constant $a_{ij}$ is hypoelliptic, translation invariant w.r.t. a Lie group operation in $\mathbb{R}^{N+1}$ and $2$-homogeneous w.r.t. a family of nonisotropic dilations. The coefficients $a_{ij}$ are bounded and Hölder continuous in space (w.r.t. some distance induced by $\mathcal{L}$ in $\mathbb{R}^{N}$) and onl…
On the kinetic Fokker--Planck equation in curved geometry
本讲义研究弯曲几何中紧致黎曼流形切丛上的动力学Fokker–Planck方程，扩展相关分析工具，在L^2和L^1框架下分析其正则化与平衡性质，统一简化并得到新结果。
Kinetic Fokker-Planck Equations with Nonlinear Diffusion
研究非线性动力学Fokker-Planck方程的存在性、正则性和唯一性，通过动力学半群平滑估计处理仅作用于速度变量的非线性扩散，建立弱解和Besov正则性，并在质量临界条件下得到唯一性。
Global Sobolev theory for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and $VMO$ in space
本文研究了具有时间可测和空间VMO系数的Kolmogorov-Fokker-Planck算子的全局Sobolev估计及其Cauchy问题的适定性。
Generic simplicity for self-adjoint operators under bounded potential perturbations
研究有界势扰动下自伴算子谱的通用简单性，通过建立抽象准则并应用于多种几何与分析设定，证明在一般势扰动下所有特征值均为单重。
We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind \[ \mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} +\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}% \] where $(a_{ij}) ,(b_{ij}) $ are constant matrices, $(a_{ij}) $ is symmetric positive definite on $\mathbb{R} ^{p_{0}}$ ($p_{0}\leq N$), and $(b_{ij}) $ is such that $\mathcal{A}$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1<p<\infty$) of the kind:% \[ \Vert \partial_{x_{i}x_{j}}^{2}u\Vert_{L^{p}(\mathbb{R}% ^{N})}\leq c\{\Vert \mathcal{A}u\Ver…
We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind [\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x) \partial_{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}%] where $(a_{ij})$ is symmetric uniformly positive definite on $\mathbb{R}^{p_{0}}$ ($p_{0}\leq N$), with uniformly continuous and bounded entries, and $(b_{ij})$ is a constant matrix such that the frozen operator $\mathcal{A}_{x_{0}}$ corresponding to $a_{ij}(x_{0})$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1<p<\infty$) of the kind:% [|\partia…
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