Oxana Manita
We obtain estimates for the Kantorovich functionals between solutions to different Fokker -- Planck -- Kolmogorov equations for measures with same diffusion part but different drifts and different initial conditions. We show possible applications of such estimates to the study of the well-posedness for nonlinear equations.
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。
继续探索
We consider Fokker--Planck--Kolmogorov equations with unbounded coefficients and obtain upper estimates of solutions. We also obtain new estimates involving Lyapunov functions.
In this article, we obtain hessian estimates for Kolmogorov-Fokker-Planck operators in non-divergence form in several Banach function spaces. Our approach relies on a representation formula and newly developed sparse domination techniques in Harmonic Analysis. Our result when restricted to weighted Lebesgue spaces yields sharp quantitative hessian estimates for the Kolmogorov-Fokker-Planck operators.
We study the Cauchy problem for Fokker--Planck--Kolmogorov equations with unbounded and degenerate coefficients. Sufficient conditions for the existence and uniqueness of solutions are indicated.
We study uniqueness of flows of probability measures solving the Cauchy problem for nonlinear Fokker-Planck-Kolmogorov equation with unbounded coefficients. Sufficient conditions for uniqueness are indicated and examples of non-uniqueness are constructed.
Analytic smoothing effect of Cauchy problem for a class of Kolmogorov-Fokker-Planck equations
本文研究一类Kolmogorov-Fokker-Planck方程的Cauchy问题，证明了在正时间内，L2初值对应的解具有解析光滑效应。
Boundedness estimates for nonlinear nonlocal kinetic Kolmogorov-Fokker-Planck equations
本文针对非线性非局部动力学Kolmogorov-Fokker-Planck方程，证明了弱下解的插值型先验有界性估计，并用尾项刻画扩散的非局部贡献。
We present global Schauder type estimates in all variables and unique solvability results in kinetic Hölder spaces for kinetic Kolmogorov-Fokker-Planck (KFP) equations. The leading coefficients are Hölder continuous in the $x, v$ variables and are merely measurable in the temporal variable. Our proof is inspired by Campanato's approach to Schauder estimates and does not rely on the estimates of the fundamental solution of the KFP operator.
In this paper we show that Markov uniqueness for symmetric pre-Dirichlet operators $L$ follows from the uniqueness of the corresponding Fokker-Planck-Kolmogorov equation (FPKE). Since in recent years a considerable number of uniqueness results for FPKE's have been achieved, we obtain new Markov uniqueness results in concrete cases. A selection of such will be presented in this paper. They include cases with killing and with degenerate diffusion coefficients.
Well-posedness of Kolmogorov-Fokker-Planck equations with unbounded drift
本文研究漂移项无界且时间仅可测、空间局部Hölder连续的Kolmogorov-Fokker-Planck方程，通过推广parametrix方法证明了Cauchy问题测度解的存在唯一性及其与随机表述的等价性。
We prove that every probability measure $μ$ satisfying the stationary Fokker-Planck-Kolmogorov equation obtained by a $μ$-integrable perturbation $v$ of the drift term $-x$ of the Ornstein-Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure $γ$ and for the density $f=dμ/dγ$ the integral of $f |\log (f+1)|^α$ against $γ$ is estimated via $\|v\|_{L^1(μ)}$ for all $α<1/4$, which is a weakened $L^1$-analog of the logarithmic Sobolev inequality. This means that stationary measures of diffusions whose drifts are integrable perturbations of $-x$ are absolute…
We present a priori estimates and unique solvability results in the mixed-norm Lebesgue spaces for kinetic Kolmogorov-Fokker-Planck (KFP) equation in divergence form. The leading coefficients are bounded uniformly nondegenerate with respect to the velocity variable $v$ and satisfy a vanishing mean oscillation (VMO) type condition. We consider the $L_2$ case separately and treat more general equations which include the relativistic KFP equation.
We consider a Kolmogorov-Fokker-Planck operator of the kind studied by Lanconelli-Polidoro in [Rend. Sem. Mat. Univ. Politec. Torino 52 (1994)], where the leading coefficients $a_{ij}$, instead of being constant, are bounded measurable functions of t. We construct an explicit fundamental solution for this operator, study its property, show a comparison result between this function and the fundamental solution of some model operators with constant $a_{ij}$, and show the unique solvability of the Cauchy problem under various assumptions on the initial datum.
We prove invariant Harnack inequalities for certain classes of non-divergence form equations of Kolmogorov type. The operators we consider exhibit invariance properties with respect to a homogeneous Lie group structure. The coefficient matrix is assumed either to satisfy a Cordes-Landis condition on the eigenvalues, or to admit a uniform modulus of continuity.
In this paper we establish optimal isoperimetric inequalities for a nonlocal perimeter adapted to the fractional powers of a class of Kolmogorov-Fokker-Planck operators which are of interest in physics. These operators are very degenerate and do not possess a variational structure. The prototypical example was introduced by Kolmogorov in his 1938 paper on brownian motion and the theory of gases. Our work has been influenced by ideas of M. Ledoux in the local case.
This work is devoted to the study of the obstacle problem associated to the Kolmogorov-Fokker-Planck operator with rough coefficients through a variational approach. In particular, after the introduction of a proper anisotropic Sobolev space and related properties, we prove the existence and uniqueness of a weak solution for the obstacle problem by adapting a classical perturbation argument to the convex functional associated to the case of our interest. Finally, we conclude this work by providing a one-sided associated variational inequality, alongside with an overview on related open problem…
A (2+1)-dimensional linear ultra-parabolic Fokker--Planck--Kolmogorov equation is investigated from the group-theoretical point of view. By using the Berest--Aksenov approach, an algebra of invariance of fundamental solutions of the equation is found. A fundamental solution of the equation under study is computed in an explicit form as a weak invariant solution.
Analysis of mean field games via Fokker-Planck-Kolmogorov equations: existence of equilibria
本文通过福克-平面克-科尔莫戈罗夫方程等方法，证明了具有无界系数的均场博弈解的存在性。
We study the degenerate Kolmogorov equations (also known as kinetic Fokker-Planck equations) in nondivergence form. The leading coefficients $a^{ij}$ are merely measurable in $t$ and satisfy the vanishing mean oscillation (VMO) condition in $x, v$ with respect to some quasi-metric. We also assume boundedness and uniform nondegeneracy of $a^{ij}$ with respect to $v$. We prove global a priori estimates in weighted mixed-norm Lebesgue spaces and solvability results. We also show an application of the main result to the Landau equation. Our proof does not rely on any kernel estimates.
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。
In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with distribution-valued inhomogeneous term, we show the existence of weak solutions under mild assumptions. Moreover, by using the Hölder regularity estimate obtained recently in \cite{GIMV19}, we also show the well-posedness of generalized martingale problems when diffusion coefficients only depend on the position variable (not necessarily continuous). Even in the non dens…
We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form \begin{equation}\label{abeqn} (\partial_t+X\cdot\nabla_Y)u=\nabla_X\cdot(A(\nabla_X u,X,Y,t)). \end{equation} The function $A=A(ξ,X,Y,t):\R^m\times\R^m\times\R^m\times\R\to\R^m$ is assumed to be continuous with respect to $ξ$, and measurable with respect to $X,Y$ and $t$. $A=A(ξ,X,Y,t)$ is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and H{ö}lder continuity with quantitative estimates. In addition we e…
Fokker-Planck-Kolmogorov inclusions of the mean field type
研究瓦瑟斯坦空间中由特定福克-普朗克-柯尔莫哥洛夫方程控制的动力系统，在多值映射满足一定条件下证明存在性定理和解集紧致性，还研究了该系统的最优控制问题。
We interpret a class of nonlinear Fokker-Planck equations with reaction as gradient flows over the space of Radon measures equipped with the recently introduced Hellinger-Kantorovich distance. The driving entropy of the gradient flow is not assumed to be geodesically convex or semi-convex. We prove new generalized dissipation inequalities, which allow us to control the relative entropy by its production. We establish the entropic exponential convergence of the trajectories of the flow to the equilibrium. Along with other applications, this result has an ecological interpretation as a trend to …
Global $L^p$ estimate for some kind of Kolmogorov-Fokker-Planck Equations in nondivergence form
本文针对具有四种不同标度且系数仅在时间上可测、空间上满足消失平均振荡条件的非散度型Kolmogorov-Fokker-Planck算子，建立了$L^p$全局先验估计，推广了已有三种标度情形，并给出齐次方程的Poincare不等式。
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 …
Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory
我们得到了非线性退化抛物方程\[ \frac{\partial ρ}{\partial t} = {div} \Big\{ρ\nabla c^\star [ \nabla (F^\prime(ρ)+V) ] \Big\} \]的解作为能量关于凸代价泛函的最速下降。这里使用的方法是变分的。它需要比Alt和Luckhaus在他们的开创性工作\cite{luckhaus:quasilinear}中所施加的一致凸性假设更少的一致凸性假设。事实上，他们的假设可能在我们的等式中失败。这类问题包括Fokker-Planck方程、多孔介质方程、快速扩散方程和抛物型p-Laplacian方程。
A study of the Kuramoto model for synchronization phenomena based on degenerate Kolmogorov-Fokker-Planck equations
该研究针对带惯性的Kuramoto同步模型对应的非线性偏微分方程，依托退化Kolmogorov算子理论证明柯西问题解的相关性质，构造相容稳定数值算子，并通过数值实验明确同步现象与模型参数的关联。
A unifying formulation of the Fokker-Planck-Kolmogorov equation for general stochastic hybrid systems
在广义随机混杂系统（GSHS）的框架下，给出了随机混杂系统的Fokker-Planck-Kolmogorov（FPK）方程的一般形式。FPK方程描述了杂化态几率律随时间的演化。我们的推导是基于平均跳跃强度的概念，这是通常的随机强度（在自发跳跃的情况下）和概率流的概念（在强迫跳跃的情况下）。这项工作统一了所有以前已知的FPK方程的随机混合系统的实例，并提供了GSHS从业者的工具，以获得正确的发展方程的概率定律的状态在任何给定的例子。
Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness
本文研究了Kolmogorov-Fokker-Planck方程弱解的存在性、唯一性和正则性，提出了一组尖锐的动能嵌入和正则性转移工具，适用于广泛正则性参数范围。
Fundamental solutions to Kolmogorov-Fokker-Planck equations with rough coefficients: existence, uniqueness, upper estimates
本文研究有限与无限时间条上带可测系数及局部或积分扩散的 Kolmogorov-Fokker-Planck 方程，证明基本解算子的存在唯一性，并建立 Gaussian 上界与 Moser 估计的等价关系。
This paper has been withdrawn from the arXiv. It is now published by Elsevier in Nonlinear Analysis: Hybrid Systems, see http://dx.doi.org/10.1016/j.nahs.2009.07.008 . A general formulation of the Fokker-Planck-Kolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHS). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on the concept of mean jump intensity, which is related to both the usual stochastic intensity (in the case of spontaneous jumps) and t…
Nonlinear Fokker--Planck--Kolmogorov equations as gradient flows on the space of probability measures
本文提出一种不依赖度量结构的微分几何方法，将非线性Fokker--Planck--Kolmogorov方程识别为概率测度空间上的梯度流，明确熵函数与能量泛函，证明唯一性及梯度场逼近，涵盖经典与广义多孔介质方程。
In this paper, we develop an operator splitting scheme for the fractional kinetic Fokker-Planck equation (FKFPE). The scheme consists of two phases: a fractional diffusion phase and a kinetic transport phase. The first phase is solved exactly using the convolution operator while the second one is solved approximately using a variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. We prove the convergence of the scheme to a weak solution to FKFPE. As a by-product of our analysis, we also establish a variational formulation …
Monge-Kantorovich distance for PDEs: the coupling method
该研究非正式回顾了多类偏微分方程（如热方程、福克-普朗克方程等），通过构建解的耦合（变量加倍）证明其解对的蒙日-康托罗维奇距离衰减，还补充了多孔介质方程的处理思路及对偶问题方法。
Convergence to equilibrium for the kinetic Fokker-Planck equation on the torus
该研究针对环面上的动理学Fokker-Planck方程，通过求解随机微分方程证明其在$\u202fW_2$距离下指数收敛至平衡态，还发现共适应耦合下的收敛界需依赖初始距离的平方根。
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。
We shall prove new contraction properties of general transportation costs along nonnegative measure-valued solutions to Fokker-Planck equations in $R^d$, when the drift is a monotone (or $λ$-monotone) operator. A new duality approach to contraction estimates has been developed: it relies on the Kantorovich dual formulation of optimal transportation problems and on a variable-doubling technique. The latter is used to derive a new comparison property of solutions of the backward Kolmogorov (or dual) equation. The advantage of this technique is twofold: it directly applies to distributional solut…
We establish the existence and uniqueness, in bounded as well as unbounded Lipschitz type cylinders of the forms $U_X\times V_{Y,t}$ and $Ω\times \mathbb R^{m}\times \mathbb R$, of weak solutions to Cauchy-Dirichlet problems for the strongly degenerate parabolic operator \[ \mathcal{L}:= \nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_Y-\partial_t, \] assuming that $A=A(X,Y,t)=\{a_{i,j}(X,Y,t)\}$ is a real $m\times m$-matrix valued, measurable function such that $A(X,Y,t)$ is symmetric, bounded and uniformly elliptic. Subsequently we solve the continuous Dirichlet problem and establish the repres…
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…
In this paper, we construct the fundamental solution to a degenerate diffusion of Kolmogorov type and develop a time-discrete variational scheme for its adjoint equation. The so-called mean squared derivative cost function plays a crucial role occurring in both the fundamental solution and the variational scheme. The latter is implemented by minimizing a free energy functional with respect to the Kantorovich optimal transport cost functional associated with the mean squared derivative cost. We establish the convergence of the scheme to the solution of the adjoint equation generalizing previous…
Infinite-dimensional nonlinear stationary Fokker-Planck-Kolmogorov equations
本文证明在无穷维空间上，具有中心高斯测度γ、单位扩散算子和形如-x+v(p,x)的漂移的非线性平稳Fokker-Planck-Kolmogorov方程存在概率解，其中v是取值于γ的Cameron-Martin空间H的有界映射，且v定义在由概率密度构成的L^2(γ)的子集E与X的乘积空间上。
We consider weak solutions of degenerate second order partial differential equations of Kolmogorov-Fokker-Planck type with measurable coefficients in divergence form. We give a geometric statement of the Harnack inequality recently proven by Golse, Imbert, Mouhot and Vasseur. As a corollary we obtain a strong maximum principle.
We consider the so-called spatially homogenous Kolmogorov-Vicsek model, a non-linear Fokker-Planck equation of self-driven stochastic particles with orientation interaction under the space-homogeneity. We prove the global existence and uniqueness of weak solutions to the equation. We also show that weak solutions exponentially converge to a steady state, which has the form of the Fisher-von Mises distribution.
Sobolev embeddings for kinetic Fokker-Planck equations
本文研究满足弱 Hörmander 条件的超抛物 Kolmogorov 型算子，构造内在 Sobolev-Slobodeckij 空间，并证明其到 Lorentz 与内在 Hölder 空间的连续嵌入及相关逼近、插值不等式。
Nash-Aronson Estimate for the Linear Kinetic Fokker-Planck equation
本文研究线性动能福克-普朗克方程的解的上界，区分长时和短时时的两种情形，长时情形得到高斯上界，短时情形则由常系数柯尔莫哥洛夫方程主导。
Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker-Planck-Kolmogorov equations
本文研究了由离开给定领域时被杀死的Feller过程生成的半群，这些半群对应于一类演化方程的Cauchy-Dirichlet型初始-外部值问题。通过Chernoff定理对这些半群进行近似，并将构造的Chernoff近似转化为Feynman公式，即n重积分的极限。此外，提出了一种近似时间分数阶（包括分布阶时间分数阶）演化方程解的方法，该方法基于时间分数阶与时间非分数阶演化方程之间的联系以及Chernoff近似。进一步，该方法导致时间分数阶演化方程解的Feynman公式。通过考虑一类分布阶时间分数阶扩散方程，获得了对应的Cauchy和Cauchy-Dirichlet问题的Feynman公式。
The Kolmogorov operator associated to a Burgers SPDE in spaces of continuous functions
本文研究受白噪声扰动的粘性Burgers方程。我们研究了相应的转移半群空间中的连续函数加权适当的潜力，我们表明，无穷小生成元是封闭的Kolmogorov算子的随机方程（关于适当的拓扑）。在本文的最后一部分，我们利用这个结果求解了相应的Fokker-Planck方程。
We prove sharp Harnack inequalities for a family of Kolmogorov-Fokker-Planck type hypoelliptic diffusions.
We study the global existence and uniqueness of weak solutions to kinetic Kolmogorov-Vicsek models that can be considered a non-local non-linear Fokker-Planck type equation describing the dynamics of individuals with orientational interactions. This model is derived from the discrete Couzin-Vicsek algorithm as mean-field limit \cite{B-C-C,D-M}, which governs the interactions of stochastic agents moving with a velocity of constant magnitude, i.e. the the corresponding velocity space for these type of Kolmogorov-Vicsek models are the unit sphere. Our analysis for $L^p$ estimates and compactness …
Generalized symmetries of remarkable (1+2)-dimensional Fokker-Planck equation
本研究用原创方法求出特殊(1+2)维Kolmogorov方程的广义对称代数，揭示其本质子代数生成机制与同构性质，并将相关理论应用于该方程的精确求解。
The aim of the present paper is twofold:(1) We carry on with developing an abstract method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators writing as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another-smaller or larger-Banach space under the condition…
A Dynamical Equation for the Lorenz Curve: Dynamics of incomplete moments of probability distributions arising from Fokker-Planck equations
本文提出洛伦兹动力学，通过将福克-普朗克方程的不完整零阶和一阶矩转化为紧凑域上的动力学，研究概率分布的动力学特性。
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…
Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence
本研究针对Fokker-Planck优化，协调了宏观与微观两种伴随视角的对应关系，证明离散化后两者梯度仍为一致近似并建立了显式收敛速率。
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。
Optimal regularity for kinetic Fokker-Planck equations in domains
该研究针对带镜面反射条件的线性动理学Fokker-Planck方程，推导了解在区域边界擦边集内外的正则性，证明了$C^{4,1}_{\text{kin}}$的最优正则性，首次证实解在边界上为经典解。
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。
We revisit the entire framework of group classification of differential equations. After introducing the notion of weakly similar classes of differential equations, we develop the mapping method of group classification for such classes, which generalizes all the versions of this method that have been presented in the literature. The mapping method is applied to group classification of various classes of Kolmogorov equations and of Fokker-Planck equations in the case of space dimension one. The equivalence groupoids and the equivalence groups of these classes are computed. The group classificat…
In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce …
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.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。