Uniform Modulus of Continuity of Random Fields

Modulus of continuity of a class of monofractal processes

We derive the modulus of continuity of a class of processes called Canonical Embedded Branching Processes (CEBP), recently introduced by Decrouez and Jones, and we establish their monofractal character. CEBP provide a rich class of processes, including the Brownian motion as a particular case. The techniques developed in this study follow the steps of Barlow and Perkins on Brownian motion on a Sierpinski gasket, though complications arise here since CEBP are not Markovian in general.

Strong Local Nondeterminism and Exact Modulus of Continuity for Spherical Gaussian Fields

In this paper, we are concerned with sample path properties of isotropic spherical Gaussian fields on $§^2$. In particular, we establish the property of strong local nondeterminism of an isotropic spherical Gaussian field based on the high-frequency behaviour of its angular power spectrum; we then exploit this result to establish an exact uniform modulus of continuity for its sample paths. We also discuss the range of values of the spectral index for which the sample functions exhibit fractal or smooth behaviour.

A stochastic calculus proof of the CLT for the L^{2} modulus of continuity of local time

We give a stochastic calculus proof of the Central Limit Theorem \[ {\int (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\Longrightarrow}c(\int (L^{x}_{t})^{2} dx)^{1/2} η\] as $h\to 0$ for Brownian local time $L^{x}_{t}$. Here $η$ is an independent normal random variable with mean zero and variance one.

On (signed) Takagi-Landsberg functions: $p^{\text{th}}$ variation, maximum, and modulus of continuity

We study a class $\mathfrak X^H$ of signed Takagi-Landsberg functions with Hurst parameter $H\in(0,1)$. We first show that the functions in $\mathfrak X^H$ admit a linear $p^{\text{th}}$ variation along the sequence of dyadic partitions of $[0,1]$, where $p=1/H$. The slope of the linear increase can be represented as the $p^{\text{th}}$ absolute moment of the infinite Bernoulli convolution with parameter $2^{H-1}$. The existence of a continuous $p^{\text{th}}$ variation enables the use of the functions in $\mathfrak X^H$ as test integrators for higher-order pathwise Itô calculus. Our next resu…

调和分数稳定运动的最优一致连续模

An Optimal Uniform Modulus of Continuity for Harmonizable Fractional Stable Motion

本文解决了非高斯调和分数稳定运动（HFSM）一致连续模对数因子最优幂次的公开问题，核心方法是对其小波展开随机系数的LePage级数做Abel变换，方法可推广至更一般的调和稳定过程与随机场。

布朗运动连续模的时间一致控制

Uniform in time modulus of continuity of Brownian motion

本文研究标准布朗运动连续模的时间一致控制，通过控制特定随机变量的指数矩，推导出扩散过程的稳定性不等式，并应用于两个简单框架。

Maximal Moments and Uniform Modulus of Continuity for Stable Random Fields

In this work, we solve an open problem mentioned in Xiao (2010) and provide sharp bounds on the rate of growth of maximal moments for stationary symmetric stable random fields using structure theorem of finitely generated abelian groups and ergodic theory of quasi-invariant group actions. We also investigate the relationship between this rate of growth and the path regularity properties of self-similar stable random fields with stationary increments, and establish uniform modulus of continuity of such fields. In the process, a new notion of weak effective dimension is introduced for stable ran…

On the Continuity of SLE(κ) in κ

We prove that for almost every Brownian motion sample, the corresponding SLE(κ) curves parameterized by capacity exist and change continuously in the supremum norm when κvaries in the interval [0,κ_0), where κ_0=8(2-\sqrt{3})=2.143... We estimate the κ-dependent modulus of continuity of the curves and also give an estimate on the modulus of continuity for the supremum norm change with κ.

A CLT for the L^{2} modulus of continuity of Brownian local time

Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion and \[ α_{t}:=\int_{-\infty}^{\infty} (L^{x}_{t})^{2} dx . \] Let $η=N(0,1)$ be independent of $α_{t}$. For each fixed $t$ \[ {\int_{-\infty}^{\infty} (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\to}({64 \over 3})^{1/2}\sqrt{α_{t}} η, \] as $h\rar 0$. Equivalently \[ {\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- 4t\over t^{3/4}} \stackrel{\mathcal{L}}{\to}({64 \over 3} )^{1/2}\sqrt{α_{1}} η, \] as $t\rar\infty$.

Exact uniform modulus of continuity and Chung's LIL for the generalized fractional Brownian motion

The generalized fractional Brownian motion (GFBM) $X:=\{X(t)\}_{t\ge0}$ with parameters $γ\in [0, 1)$ and $α\in \left(-\frac12+\fracγ{2}, \, \frac12+\fracγ{2} \right)$ is a centered Gaussian $H$-self-similar process introduced by Pang and Taqqu (2019) as the scaling limit of power-law shot noise processes, where $H = α-\fracγ{2}+\frac12 \in(0,1)$. When $γ= 0$, $X$ is the ordinary fractional Brownian motion. For $γ\in (0, 1)$, GFBM $X$ does not have stationary increments, and its sample path properties such as Hölder continuity, path differentiability/non-differentiability, and the functional l…

Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic Gaussian Random Fields on Compact Two-Point Homogeneous Spaces

This paper is concerned with sample path properties of isotropic Gaussian fields on compact two-point homogeneous spaces. In particular, we establish the property of strong local nondeterminism of an isotropic Gaussian field based on the high-frequency behavior of its angular power spectrum, and then exploit this result to establish an exact uniform modulus of continuity for its sample paths.

分数布朗运动：具有改进的几乎必然上界的局部连续模及从单侧边界首次退出时间

Fractional Brownian Motion: Local Modulus of Continuity with Refined Almost Sure Upper Bound and First Exit Time from One-sided Barrier

本文基于最优速率小波级数表示，推导出分数布朗运动局部连续模的改进几乎必然上界及其 $p$ 阶矩上界，填补了迭代对数律的空白，并据此给出了从正值边界首次退出时间的改进上尾概率渐近估计。

Factorable continuity of random fields, with quantitative estimation

We study in this paper the sufficient conditions for enhanced continuity of random fields, i.e. such that the modulus of its continuity allows the factorable representation by the product of random variable on the deterministic module of continuity. We estimate also the ordinary and (possible) exponential moments of these random variables. We consider also the case of random fields with heavy tails of distribution and the so-called rectangle its continuity.

连续Young图的Kerov转移测度的连续性模

Modulus of continuity of Kerov transition measure for continual Young diagrams

本文量化了连续Young图子类与中心化概率测度类之间Kerov转移测度同胚的连续性模，重点研究了累积分布函数在轮廓不太陡峭处对图轮廓的依赖性。

Modulus of Continuity of Controlled Loewner-Kufarev Equations and Random Matrices

First we introduce the two tau-functions which appeared either as the $τ$-function of the integrable hierarchy governing the Riemann mapping of Jordan curves or in conformal field theory and the universal Grassmannian. Then we discuss various aspects of their interrelation. Subsequently, we establish a novel connection between free probability, growth models and integrable systems, in particular for second order freeness, and summarise it in a dictionary. This extends the previous link between conformal maps and large $N$-matrix integrals to (higher) order free probability. Within this context…

Continuity estimates for the complex cascade model on the phase boundary

We consider the complex branching random walk on a dyadic tree with Gaussian weights on the boundary between the diffuse phase and the glassy phase. We study the branching random walk in the space of continuous functions and establish convergence in this space. The main difficulty here is that the expected modulus of continuity of the limit is too weak in order to show tightness in the space of continuous functions by means of standard tools from the theory of stochastic processes.

SLE的正则性：至常数变化和连续模

Regularity of the Schramm-Loewner evolution: Up-to-constant variation and modulus of continuity

本文研究了SLE的正则性，确定了至常数变化和连续模的最优界，并证明了自然参数化由ψ-变化的细网格极限给出。

Moduli of continuity of local times of random walks on graphs in terms of the resistance metric

In this article, universal concentration estimates are established for the local times of random walks on weighted graphs in terms of the resistance metric. As a particular application of these, a modulus of continuity for local times is provided in the case when the graphs in question satisfy a certain volume growth condition with respect to the resistance metric. Moreover, it is explained how these results can be applied to self-similar fractals, for which they are shown to be useful for deriving scaling limits for local times and asymptotic bounds for the cover time distribution.

Modulus of continuity for polymer fluctuations and weight profiles in Poissonian last passage percolation

In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics. The geodesics and their energy can be scaled so that transformed geodesics cross unit distance and have fluctuations and scaled energy of unit order. Here we consider Poissonian last passage percolation, a model lying in the KPZ universality class, and refer to scaled geodesics as polymers and their scaled energies as weights. Polymers may be viewed as random functions of the vertical coordinate and, when they…

Modulus of continuity of polymer weight profiles in Brownian last passage percolation

In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths' pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, one endpoint of such polymers is fixed, say at $(0,0) \in \mathbb{R}^2$, and the other is varied horizontall…

Absolute Continuity and Weak Uniform Mixing of Random Walk in Dynamic Random Environment

We prove results for random walks in dynamic random environments which do not require the strong uniform mixing assumptions present in the literature. We focus on the "environment seen from the walker"-process and in particular its invariant law. Under general conditions it exists and is mutually absolutely continuous to the environment law. With stronger assumptions we obtain for example uniform control on the density or a quenched CLT. The general conditions are made more explicit by looking at hidden Markov models or Markov chains as environment and by providing simple examples.

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。

随机场的博雷尔变换的分布一致连续性

Uniform Continuity in Distribution for Borel Transformations of Random Fields

本文提供了随机场博雷尔变换分布一致连续性的简单充分条件。

与非一致扩张圆周映射相关的平衡态关联指数衰减速率

Exponential rate of decay of correlations of equilibrium states associated with non-uniformly expanding circle maps

本文研究带无差别不动点的非一致扩张圆周映射，证明当连续性模Ω满足特定极限条件时，系统关于以ω为连续性模的势函数所对应的唯一平衡态具有关联指数衰减，其核心在于转移算子的谱隙性质，并应用于Manneville-Pomeau族。

On uniform continuity of posterior distributions

In the setting of dominated statistical models, we provide conditions yielding strong continuity of the posterior distribution with respect to the observed data. We show some applications, with special focus on exponential models.

关于截断高斯、乘积柯西、玻尔兹曼及Softmax策略的均匀策略得分界与利普希茨连续性的注记

Notes on Uniform Policy Score Bounds and Lipschitz Continuity for Truncated Gaussian, Product Cauchy, Boltzmann, and Softmax Policies

该研究针对截断高斯、乘积柯西、玻尔兹曼及Softmax策略，在明确参数化下建立了均匀策略得分界与利普希茨连续性。

Uniform continuity bound for sandwiched Rényi conditional entropy

We prove a simple uniform continuity bound for the sandwiched Rényi conditional entropy for $α\in [1/2, 1) \cup (1, \infty]$, which is independent of the dimension of the conditioning system.