两类分支模型的重整化变换

On the empty balls of a critical or subcritical branching random walk

Let $\{Z_n\}_{n\geq 0 }$ be a critical or subcritical $d$-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on $\mathbb{R}^d$. Denote by $R_n:=\sup\{u>0:Z_n(\{x\in\mathbb{R}^d:|x|<u\})=0\}$ the radius of the largest empty ball centered at the origin of $Z_n$. In this work, we prove that after suitable renormalization, $R_n$ converges in law to some non-degenerate distribution as $n\to\infty$. Furthermore, our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random wal…

通过分支繁殖环境变换的离散稳定过程

Transformation of the discrete stable process via branching reproduction environment

本文研究无限均值移民进入无限方差繁殖环境的影响，推导概率生成函数显式形式并分析灭绝概率，以理解长期行为。

关于连续状态分支过程：条件作用和自相似性

On continuous state branching processes: conditioning and self-similarity

本文证明了$α\in (1, 2}$的$α$ -稳定的连续态分支过程及其相伴的永不灭绝的过程是正自相似马氏过程.理解连续状态分支过程的Lamperti变换和正自相似Markov过程的Lamperti变换的相互作用，可以获得一些关于稳定连续分支过程及其条件版本的路径的明确结果。

重整化群变换的马尔可夫链方法

A Markov chain approach to renormalization group transformations

我们的目标是明确的重整化哈密顿量的抽取变换后的最近邻相互作用和磁场项的一维伊辛型哈密顿量的表征。为了便于更深入地理解抽取效应，我们通过Markov-Gibbs等价将伊辛哈密顿量上的重整化流转化为相关马尔可夫链上的流。用两种不同的方法来验证著名的猜想，即关于不动点的重整化变换的线性化的特征值具有关于所有六个临界指数的重要信息。这说明了这种情况下的重整化群映射的普适性。

On multitype Branching Processes with Interaction

Motivated by the stochastic Lotka-Volterra model, we introduce discrete-state interacting multitype branching processes. We show that they can be obtained as the sum of a multidimensional random walk with a Lamperti-type change proportional to the population size; and a multidimensional Poisson process with a time-change proportional to the pairwise interactions. We define the analogous continuous-state process as the unique strong solution of a multidimensional SDE. We prove that the scaling limits of the discrete-state process correspond to its continuous counterpart. In addition, we show th…

竞争分枝过程与广义Ray Knight定理

Branching processes with competition and generalized Ray Knight Theorem

本文考虑一个具有相互作用的离散人口模型，其中出生率和死亡率是总体规模的非线性函数。在对模型参数进行重整化之后，我们得到在大布居极限下，总体规模演化为下式的扩散解Z^x_t =x+ \int_0^t f（Z^x_s）ds+2 \int_0^t \int_0^{Z^x_s} W（ds，du），其中W（ds，du）是（[0，\infty））^2上的时空白噪声。我们给出了一个Ray-Knight表示的这种扩散的反射布朗运动H的本地时间的漂移，取决于本地时间积累的H在其当前的水平，通过函数f '/2。