Characterization of polynomials

超临界 Bernoulli 渗流的扩散系数是无限可微的

The diffusivity of supercritical Bernoulli percolation is infinitely differentiable

本文证明超临界 $\mathbb{Z}^d$-Bernoulli 渗流的扩散系数与电导率无限可微，并通过受扰校正方程、多尺度几何重整化及新分解技术推广 Kozlov 的结果。

Ladder Theorem for a Leray alpha model of turbulence

In this paper, we study the Modified Leray alpha model with periodic boundary conditions. We show that when the initial data are infinitely differentiable then the unique solution are infinitely differentiable in space and time. Furthermore, this regular solution verifies a sequence of energy inequalities that is called "ladder inequalities".

Hörmander's theorem for parabolic equations with coefficients measurable in the time variable

We are dealing with possibly degenerate second-order parabolic operators whose coefficients are infinitely differentiable with respect to space variables and only measurable with respect to the time variable. We impose the Hörmander condition on the diffusion coefficients and prove that the solutions of the corresponding equations with right-hand sides which are infinitely differentiable in the space variables in a space-time domain have also this property.

Hypoellipticity for filtering problems of partially observable diffusion processes

We prove that under Hörmander's type conditions on the coefficients of the unobservable component of a partially observable diffusion process the filtering density is infinitely differentiable and can be represented as the integral of an infinitely differentiable kernel against the prior initial distribution. These results are derived from more general results obtained for SPDEs. One the main novelty of the paper is the existence and smoothness of the kernel, another that we allow the coefficients of our partially observable process to be just measurable with respect to the time variable.

Construction of solutions to parabolic and hyperbolic initial-boundary value problems

We show that infinitely differentiable solutions to parabolic and hyperbolic equations, whose right-hand sides are analytical in time, are also analytical in time at each fixed point of the space. These solutions are given in the form of the Taylor expansion with respect to time $t$ with coefficients depending on $x$. The coefficients of the expansion are defined by recursion relations, which are obtained from the condition of compatibility of order $k=\infty$. The value of the solution on the boundary is defined by the right-hand side and initial data, so that it is not prescribed. We show th…

关于Wasserstein空间中的光滑逼近

On smooth approximations in the Wasserstein space

本文在Wasserstein空间上利用Lions可微性构造光滑逼近，证明Lipschitz函数可由具有相同Lipschitz常数的无穷次可微函数列逼近，并推广到一阶和二阶连续可微情形，解决了文献[11]中的两个公开问题。