Xicheng Zhang
By using the probability approach (the Malliavin calculus), we prove the existence of smooth fundamental solutions for degenerate kinetic Fokker-Planck equation with anisotropic nonlocal dissipativity, where the dissipative term is the generator of an anisotropic Lévy process and the drift term is allowed to be cubic growth.
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A uniform-in-time nonlocal approximation of the standard Fokker-Planck equation
本文提出了一种时间一致的非局部近似方法，用于研究福克-普朗克方程的收敛性，利用哈里斯定理证明了解的自适应保持性，并分析了平衡态的正则性性质。
Nonlocal, nonlinear Fokker-Planck equations and nonlinear martingale problems
研究具有非局部算子Ψ(-Δ)的非线性Fokker-Planck方程的温和解存在性与分布解唯一性，并证明对应非线性鞅问题的解存在唯一，进而构造McKean意义下的非线性马尔可夫过程。
Time Asymptotics and Scaling Limits for a Nonlocal Fokker-Planck Equation with Heavy-Tailed Kernel
本文研究了非局部福克-普朗克方程在重尾核下的时间渐近性与标度极限，证明了解向平衡状态的指数收敛性，并展示了在参数趋于零或一的情况下的一致收敛性。
We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behaviour of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker-Planck framework, and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to …
Optimal control problems driven by nonlinear degenerate Fokker-Planck equations
本文研究了耦合非线性退化Fokker-Planck方程与常微分方程系统的最优控制问题的适定性，结合测度解稳定性估计与成本泛函假设框架确保紧性和下半连续性，利用李结构支持非Lipschitz非线性项，为多种群动力学平均场极限调控提供理论支撑。
In this paper, we study the long-time behaviour of solutions to the Vlasov-Fokker-Planck equation where the confining potential is non-convex. This is a nonlocal nonlinear partial differential equation describing the time evolution of the probability distribution of a particle moving under the influence of a non-convex potential, an interaction potential, a friction force and a stochastic force. Using the free-energy approach, we show that under suitable assumptions solutions of the Vlasov-Fokker-Planck equation converge to an invariant probability.
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 consider a nonlinear Fokker-Planck equation driven by a deterministic rough path which describes the conditional probability of a McKean-Vlasov diffusion with "common" noise. To study the equation we build a self-contained framework of non-linear rough integration theory which we use to study McKean-Vlasov equations perturbed by rough paths. We construct an appropriate notion of solution of the corresponding Fokker-Planck equation and prove well-posedness.
Stochastic differential equations with Levy motion arise the mathematical models for various phenomenon in geophysical and biochemical sciences. The Fokker Planck equation for such a stochastic differential equations is a nonlocal partial differential equations. We prove the existence and uniqueness of the weak solution for this equation.
The Fokker-Planck equations (FPEs) for stochastic systems driven by additive symmetric $α$-stable noises may not adequately describe the time evolution for the probability densities of solution paths in some practical applications, such as hydrodynamical systems, porous media, and composite materials. As a continuation of previous works on additive case, the FPEs for stochastic dynamical systems with multiplicative symmetric $α$-stable noises are derived by the adjoint operator method, which satisfy the nonlocal partial differential equations. A finite difference method for solving the nonloca…
Nonlinear rough Fokker-Planck equations
针对含共同噪声的非线性随机 Fokker–Planck 方程适定性难题，本文利用粗糙路径技术消除 Coghi–Gess 结果中依赖维数的正则性假设。
Optimal Control of the Nonlinear Stochastic Fokker--Planck Equation
本文针对非线性随机Fokker--Planck方程提出最优控制框架，证明解的存在性与最优控制存在性，推导必要且充分的随机最大值原理，并应用于政府干预金融系统模型。
Fokker-Planck equation for stochastic heat equations
该研究针对带加性Q-Wiener噪声的非齐次边界随机热方程，用本征函数展开构造概率密度函数并建立Fokker-Planck方程，结合Feynman-Kac公式得到解的概率表示，还拓展至乘性噪声非局部扩散情形及KPZ方程，发现概率密度演化关键依赖空间位置。
Kinetic Theory with Fluctuations: Well-Posedness of The Vlasov--Fokker--Planck--Dean--Kawasaki Equations
研究含相关噪声的 Vlasov-Fokker-Planck-Dean-Kawasaki 方程的强适定性，通过动理学半群估计和重整化动理学解框架，克服了动理学算子复杂性和保守噪声带来的不规则性。
Decay rates of convergence for Fokker-Planck equations with confining drift
本文针对由Levy过程驱动且带约束漂移的Fokker-Planck方程，提出一种新的PDE方法，基于伴随问题的长时间振荡估计，统一得到零平均解的指数或次指数衰减率，并建立了与Hamilton-Jacobi方程方法及概率耦合方法的联系。
Interacting particle approximation of cross-diffusion systems
本文通过非线性扩散项的非局部逼近、耦合方法和紧性论证，证明了一类多物种交叉扩散系统弱解的存在性与混沌传播，并在迁移率的结构假设下结合粘性多孔介质方程和线性 Fokker-Planck 方程证明了唯一性，同时证明了其对随机相互作用粒子系统宏观行为的刻画。
In this work, by using the Malliavin calculus, under Hörmander's condition, we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated Brownian motions. Moreover, in a special degenerate case, we also obtain the smoothness of the density. In particular, we obtain the existence of smooth heat kernels for the following fractional kinetic Fokker-Planck (nonlocal) operator: \[\mathscr{L}^{(α)}_b:=Δ^{α/2}_{\mathrm{v}}+\mathrm {v}\cdot \nabla_x+b(x,\mathrm{v})\cdot \nabla_{\mathrm{v}},\qquad x,\mathrm{v}\in\mathbb{R}^d,…
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