A limit theorem for the maximal interpoint distance of a random sample in the unit ball

Maximal displacement in a branching random walk through interfaces

In this article, we study a branching random walk in an environment which depends on the time. This time-inhomogeneous environment consists of a sequence of macroscopic time intervals, in each of which the law of reproduction remains constant. We prove that the asymptotic behaviour of the maximal displacement in this process consists of a first ballistic order, given by the solution of an optimization problem under constraints, a negative logarithmic correction, plus stochastically bounded fluctuations.

停止占用问题中的最大计数

Maximal Counts in the Stopped Occupancy Problem

本文研究停止占用问题中最大计数的渐近分布，证明其不收敛但可由两个 Gumbel 分布的卷积近似，并利用格点过程与多重集插值两种方法深入解析了其渐近性质及矩与并列情况。

Maximal displacement of a branching random walk in time-inhomogeneous environment

Consider a branching random walk evolving in a macroscopic time-inhomogeneous environment, that scales with the length $n$ of the process under study. We compute the first two terms of the asymptotic of the maximal displacement at time $n$. The coefficient of the first (ballistic) order is obtained as the solution of an optimization problem, while the second term, of order $n^{1/3}$, comes from time-inhomogeneous random walk estimates, that may be of independent interest. This result partially answers a conjecture of Fang and Zeitouni. Same techniques are used to obtain the asymptotic of other…

Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points

We show that the random point measures induced by vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled and centered, converge to those of a mean zero Gaussian field. We establish limiting variance and covariance asymptotics in terms of the density of the Poisson sample. Similar results hold for the point measures induced by the maximal points in a Poisson sample. The approach involves introducing a generalized spatial birth growth process allowing for cell overlap.

Random ballistic growth and diffusion in symmetric spaces

Sequential ballistic deposition (BD) with next-nearest-neighbor (NNN) interactions in a N-column box is viewed a time-ordered product of N\times N-matrices consisting of a single sl_2-block which has a random position along the diagonal. We relate the uniform BD growth with the diffusion in the symmetric space H_N=SL(N,R)/SO(N). In particular, the distribution of the maximal height of a growing heap is connected with the distribution of the maximal distance for the diffusion process in H_N. The coordinates of H_N are interpreted as the coordinates of particles of the one--dimensional Toda chai…

开球上带分数阶拉普拉斯算子的非线性椭圆型偏微分方程解的存在性

Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls

本文利用(2s)-稳定分支过程的树状概率表示，在开球上证明了带分数阶拉普拉斯算子的半线性椭圆型偏微分方程粘性解的存在性，并放宽了对多项式非线性次数和项数的限制。

与对数凹密度相关的极大函数的高维极限和极值点

High-dimensional limits and extremizers for maximal functions associated with log-concave densities

研究与径向对数凹概率密度相关极大函数的高维渐近行为，通过一般转移原理，利用傅里叶乘子符号及新工具给出量化界，证明欧几里得球相关极大算子有渐近最小\(L^p\)算子范数。