D. Kinzebulatov, Yu. A. Semenov
We establish sharp upper and lower bounds on the heat kernel of the fractional Laplace operator perturbed by Hardy-type drift by transferring it to appropriate weighted space with singular weight.
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We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical-order singularity, by transferring it to appropriate weighted space with singular weight.
We establish sharp two-sided weighted bounds on the fundamental solution to the fractional Schrödinger operator using the method of desingularizing weights.
Intrinsic Hardy inequality for fractional Kolmogorov operator
本文研究了当漂移项对原点的吸引力强度取临界值时，非局部扩散算子的Hardy不等式。
Fractional Kolmogorov equations with singular paracontrolled terminal conditions
本文研究带低正则奇异 Besov 漂移与奇异终端条件的倒向分数阶 Kolmogorov 方程，通过增强假设和仿控终端条件构建统一解理论。
Kinetic maximal $L^2$-regularity for the (fractional) Kolmogorov equation
该研究引入带时间权重的动力学极大$L^2$正则性，用分数阶各向异性Sobolev空间刻画（分数阶）Kolmogorov方程解的正则性，并证明齐次方程解构成半流且具有瞬时正则化性质。
Pointwise estimates of the fundamental solution to the fractional Kolmogorov equation
本研究利用傅里叶方法得到分数阶Kolmogorov方程基本解的精确双边估计，并给出拉普拉斯算子平方根情形下基本解的显式形式。
A stochastic solution is constructed for a fractional generalization of the KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution uses a fractional generalization of the branching exponential process and propagation processes which are spectral integrals of Levy processes
Kinetic maximal $L^p_μ(L^p)$-regularity for the fractional Kolmogorov equation with variable density
本文研究带可变密度的非局部分数阶 Kolmogorov 方程，证明其动力学极大 $L^p_\mu$-正则性，并用于建立拟线性分数阶动力学 PDE 强解的短时间存在性。
We study Kolmogorov's two-equation model of turbulence on $d-$dimensional torus. First, the local existence of the solution with the initial data from non-homogeneous fractional Sobolev spaces (Bessel potential spaces) $H^s$ with $s>\frac{d}{2}$ is proven using energy methods. Next, we show that solutions are unique in the class of solutions guaranteed by the local existence theorem.
We prove regularization properties in short time for inhomogeneous kinetic equations whose collision kernel behaves like a fractional power of the Laplacian in velocity. We treat a fractional Kolmogorov equation and the linearized Boltzmann equation without cutoff (for hard potentials).
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。
Variational inequalities associated with the semigroups generated by fractional Kolmogorov operators
本文研究分数阶Kolmogorov算子生成的全纯半群对应的变分算子，建立了其$L^p$有界性条件，分析了端点空间行为并证明了$V_2$算子的无界性。
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.
Let $z = (x,y) \in {\mathbb R}^d \times {\mathbb R}^{N-d}$, with $1 \le d < N$. We prove a priori estimates of the following type :$$\|Δ\_{x}^{\frac α2} v \|\_{L^p({\mathbb R}^N)} \lec\_p\Big \| L\_{x } v + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}} v \Big \|\_{L^p({\mathbb R}^N)}, \;\; 1<p<\infty,$$for $v \in C\_0^{\infty}({\mathbb R}^N)$,where $L\_x$ is a non-local operator comparable with the ${\mathbb R}^d $-fractional Laplacian $Δ\_{x}^{\frac α2}$ in terms of symbols, $α\in (0,2)$. We require that when $L\_x$ is replaced by the classical ${\mathbb R}^d$-Laplacian $Δ\_{x}$, i.e., in …
Equivalence of Sobolev norms for Kolmogorov operators with scaling-critical drift
本文研究了Kolmogorov算子中具有缩放临界漂移项的Sobolev范数等价性，指出需对可比Sobolev空间范围施加额外限制，因相关半群缺乏梯度界。
We consider the equation $(\partial_t + ρ(\sqrt{-Δ}))f(t,x) = \mathbf 1_ωu(t,x)$, $x\in \mathbb R$ or $\mathbb T$. We prove it is not null-controllable if $ρ$ is analytic on a conic neighborhood of $\mathbb R_+$ and $ρ(ξ) = o(|ξ|)$. The proof relies essentially on geometric optics, i.e.\ estimates for the evolution of semiclassical coherent states. The method also applies to other equations. The most interesting example might be the Kolmogorov-type equation $(\partial_t -\partial_v^2 + v^2\partial_x)f(t,x,v) = \mathbf 1_ωu(t,x,v)$ for $(x,v)\in Ω_x\times Ω_v$ with $Ω_x = \mathbb R$ or $\mathbb…
Fractional Sobolev spaces related to an ultraparabolic operator
本文为满足弱 Hörmander 条件的超抛物 Kolmogorov 型算子构建分数阶 Sobolev 空间框架，通过实插值与内蕴 Taylor 余项估计证明其到 L^p 和内蕴 Hölder 空间的连续嵌入。
New link between the fractional p-Laplacian operators and a class of McKean-Vlasov flight type processes
本文证明了一类与非线性抛物方程（含p-Laplacian和分数阶p-Laplacian）相关联的带跳McKean-Vlasov过程的存在性，通过重写为FPKE并利用新的非线性叠加原理解决鞅问题，最终得到弱解。
Subordinators and generalized heat kernels: Random time change and long time dynamics
本文研究线性演化方程（特别是分数阶热方程）从属过程的长时间动力学，通过逆从属子作为随机时间变量并利用从属原理求解前向Kolmogorov方程，扩展并增强了已有结果。
Kinetic maximal $L^p$-regularity with temporal weights and application to quasilinear kinetic diffusion equations
本文提出带时间权重的动力学极大 $L^p$-正则性理论，证明其适用于分数阶及变系数 Kolmogorov 方程，并用于建立一类拟线性动力学方程解的局部存在性。
Weighted Kolmogorov equations: transfer estimates, local boundedness, and Harnack inequalities
本文研究加权Kolmogorov方程，在一般A2权重下证明传递估计、局部有界性、Harnack不等式及强最小值原理，并涵盖分数幂扩展中的特定权重。
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