New existence theorems in measure theory and equivalence results for the existence of invariant probabilities

Transition probabilities of Lévy-type processes: Parametrix construction

We present an existence result for Lévy-type processes which requires only weak regularity assumptions on the symbol $q(x,ξ)$ with respect to the space variable $x$. Applications range from existence and uniqueness results for Lévy-driven SDEs with Hölder continuous coefficients to existence results for stable-like processes and Lévy-type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of Lévy and Lévy-type processes. The paper includes an extensive list of Lévy(-type) processes satisfying the assumptions of our results.

具有均匀边际的整数分拆增长过程与分拆系综的等价性

Growing integer partitions with uniform marginals and the equivalence of partition ensembles

本文构造了整数分拆上具有均匀边际的马尔可夫增长过程，分析其跳跃概率的渐近行为以解释极限形状的形成，并利用该过程给出两个随机分拆系综等价的充分条件，进而证明奇偶部分计数渐近独立且极限分布不同。

正方形晶格上临界渗流交叉概率的标度极限

A scaling limit of crossing probabilities for critical percolation on the square lattice

本文证明了正方形晶格上临界渗流在等边三角形中的交叉概率存在标度极限，并指出Cardy公式在该情形下不成立。

Intermediate spaces, Gaussian probabilities and exponential tightness

Let us consider a Gaussian probability on a Banach space. We prove the existence of an intermediate Banach space between the space where the Gaussian measure lives and its RKHS. Such a space has full probability and a compact embedding. This extends what happens with Wiener measure, where the intermediate space can be chosen as a space of Hölder paths. From this result it is very simple to deduce a result of exponential tightness for Gaussian probabilities.

临界区域中非存在概率和下尾问题的Belief Propagation方法

Non-existence probabilities and lower tails in the critical regime via Belief Propagation

本文通过Belief Propagation方法研究了临界区域中多种组合问题的非存在概率和下尾概率的对数渐进行为，适用于随机图子图和整数集中的k项等差数列。

Doob equivalence and non-commutative peaking for Markov chains

In this paper we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the non-commutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matri…

上概率的有限遍历分量

Finite ergodic components for upper probabilities

该研究在上概率遍历性框架下提出有限遍历分量（FEC）定义，证明其与不变集容度二分性、Koopman算子特征值1有限重数等价，还建立了多值大数定律等与FEC的等价性并推广至非不变概率。

一个计算概率的变分问题

A variational problem to calculate probabilities

本文通过Riesz表示定理，将条件期望的存在唯一性转化为Lebesgue空间中的线性问题，并推广为变分问题，最终用此工具证明全概率定律。

On the equivalence between some jumping SDEs with rough coefficients and some non-local PDEs

We study some jumping SDE and the corresponding Fokker-Planck (or Kolmogorov forward) equation, which is a non-local PDE. We assume only some measurability and growth conditions on the coefficients. We prove that for any weak solution $(f_t)_{t\in [0,T]}$ of the PDE, there exists a weak solution to the SDE of which the time marginals are given by $(f_t)_{t\in[0,T]}$. As a corollary, we deduce that for any given initial condition, existence for the PDE is equivalent to weak existence for the SDE and uniqueness in law for the SDE implies uniqueness for the PDE. This extends some ideas of Figalli…