Extremal solutions for stochastic equations indexed by negative integers and taking values in compact groups

多索引和高维中的多重求和

Set-indexed and multiple sums in high dimensions

研究高维空间中随机向量的多重和与集合索引和的收敛性，证明其作为有限度量空间在概率意义下收敛，极限为 Weiner 螺旋的推广。

Stable random fields indexed by finitely generated free groups

In this work, we investigate the extremal behaviour of left-stationary symmetric $α$-stable (S$α$S) random fields indexed by finitely generated free groups. We begin by studying the rate of growth of a sequence of partial maxima obtained by varying the indexing parameter of the field over balls of increasing size. This leads to a phase-transition that depends on the ergodic properties of the underlying nonsingular action of the free group but is different from what happens in the case of S$α$S random fields indexed by $\mathbb{Z}^d$. The presence of this new dichotomy is confirmed by the study…

Stable Random Fields, Patterson-Sullivan measures and Extremal Cocycle Growth

We study extreme values of group-indexed stable random fields for discrete groups $G$ acting geometrically on spaces $X$ in the following cases: 1) $G$ acts freely, properly discontinuously by isometries on a CAT(-1) space $X$, 2) $G$ is a lattice in a higher rank Lie group, acting on a symmetric space $X$, 3) $G$ is the mapping class group of a surface acting on its Teichmuller space. The connection between extreme values and the geometric action is mediated by the action of the group $G$ on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value the…

参数指标随机积分的精确上确界与Hölder界

Sharp supremum and Hölder bounds for stochastic integrals indexed by a parameter

本文利用Kolmogorov型链式论证，为2-光滑Banach空间中随机卷积的上确界及参数化随机积分的连续模提供了精确界，建立了任意有界子集上的Hölder空间随机积分理论，并应用于Ornstein-Uhlenbeck过程的长期界和抛物Anderson模型的连续模研究。

某些Banach空间中Dedecker-Rio条件下的中心极限定理

Central limit theorem under the Dedecker-Rio condition in some Banach spaces

本研究将Dedecker-Rio条件下的中心极限定理推广到取值于一类光滑Banach空间的适应平稳遍历随机变量序列，适用于$L^p(μ)$空间情形，给出Sobolev球索引经验过程满足中心极限定理的充分条件并讨论其最优性。

具有跳跃和移动边界的多值反向随机微分方程

Multivalued backward stochastic differential equations with jumps and moving boundary

本文研究了一维具有跳跃和移动边界的多值反向随机微分方程的存在性和唯一性，通过罚方法和单调性条件等证明了其解的存在性，并讨论了无界区间情况的扩展。