CS12: Recent Advances in Non-Markovian Processes and Random Fields

Organizer: Cheuk Yin Lee (The Chinese University of Hong Kong, Shenzhen)

Sample path properties of Gaussian random fields with slowly varying increments

Yimin Xiao

Let \(X_0= \{X_0(t), t \in {\mathbb R}^N\}\) be a real-valued Gaussian random field whose incremental variance is slowly varying and approximately isotropic. Let \(X=\{X(t), t\in {\mathbb R}^N\}\) be an \((N,d)\)-Gaussian random field defined by \(X(t)=(X_1(t),\ldots,X_d(t)),\) where \(X_1,\ldots, X_d\) are independent copies of \(X_0.\) In this paper, we study the exact local and uniform moduli of continuity, small ball probability estimates, Hausdorff measure of the sample paths, and tail probability estimates for the local times of the Gaussian random field \(X\). As examples, we show that our results are not only applicable to a class of Gaussian random fields with stationary increments, but also to the solution of the stochastic heat equation in Herrell et al (2020), the fractional Brownian motion of order 0 in Fyodorov et al (2016), and the logarithmic Brownian motion introduced in Mocioalca and Viens (2005).

This is a joint paper with Antoine Ayache and Dongsheng Wu.

Fourier dimension of the graph of fractional Brownian motion with H>1/2

Cheuk Yin Lee

Title: Fourier dimension of the graph of fractional Brownian motion with \(H>1/2\)
Speaker: Cheuk Yin Lee (The Chinese University of Hong Kong, Shenzhen)
Abstract: We prove that the Fourier dimension of the graph of fractional Brownian motion with Hurst index greater than 1/2 is almost surely 1. This extends the result of Fraser and Sahlsten (2018) for the Brownian motion and verifies partly the conjecture of Fraser, Orponen and Sahlsten (2014). We introduce a combinatorial integration by parts formula to compute the moments of the Fourier transform of the graph measure. The proof of our main result is based on this integration by parts formula together with Faà di Bruno’s formula and strong local nondeterminism of fractional Brownian motion. We also show that the Fourier dimension of the graph of a symmetric \(\alpha\)-stable process with \(\alpha\in[1,2]\) is almost surely 1. This is joint work with Chun-Kit Lai (San Francisco State University).

Scaling limit of dependent random walks

Jeonghwa Lee

Recently, a generalized Bernoulli process (GBP) was developed as a stationary binary sequence that can have long-range dependence, and it was further broadened to include various covariance functions. In this work, we find the scaling limits of dependent random walks that follow GBPs and the dependence structure of the limiting processes. We show that the second-type Mittag-Leffler process and exponential process arise as the limiting processes. Compound processes are considered with a Levy process subordinated to the cumulative sum in GBP, and the asymptotic properties and dependence structure of the subordinated processes are discussed. Applications of GBP are provided with datasets in economics and horticulture.

Bibliography

\([1]\) Lee, J. “Generalized Bernoulli process with long-range dependence and fractional binomial distribution." Dependence Modeling, 9, 2021, 1-12.

\([2]\) Lee, J. “Generalized Bernoulli process and fractional Poisson process." to appear in Stochastics.

\([3]\)Lee, J. “Scaling limit of dependent random walks." arXiv, 2025, arXiv:2504.14447