Abstracts

Invited Speakers

Hong-Bin Chen
Institut des Hautes Etudes Scientifiques

On Free Energy in Non-convex Mean-Field Spin Glass Models
We start by reviewing the classical Sherrington-Kirkpatrick (SK) model. In this model, +1/-1-valued spins interact with each other subject to random coupling constants. The covariance of the random interaction is quadratic in terms of spin overlaps. Parisi proposed the celebrated variational formula for the limit of free energy of the SK model in the 80s, which was later rigorously verified in the works by Guerra and Talagrand. This formula has been generalized in various settings, for instance, to vector-valued spins, by Panchenko. However, in these cases, the convexity of the interaction is crucial. In general, the limit of free energy in non-convex models is not known and we do not have variational formulas as valid candidates. Here, we report recent progress through the lens of the Hamilton-Jacobi equation. Under the assumption that the limit of free energy exists, we show that the value of the limit is prescribed by a characteristic line; and the limit (as a function) satisfies an infinite-dimensional Hamilton-Jacobi equation "almost everywhere". This talk is based on a joint work with Jean-Christophe Mourrat.

Le Chen
University of Auburn

Moment Growth and Intermittency for SPDEs in the Sublinear-growth Regime
In this paper, we investigate stochastic heat equation with sublinear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish non-trivial moment upper bounds for the solution. These moment bounds shed light on the smoothing intermittency effect under weak diffusion (i.e., sublinear growth) previously observed by Zeldovich et al. The method we employ is highly robust and can be extended to a wide range of stochastic partial differential equations, including the one-dimensional stochastic wave equation. This work is a collaboration with Panqiu Xia, formerly a postdoctoral researcher at Auburn University and now a lecturer at the University of Cardiff, UK. The preprint is available on arXiv (2306.06761) and is currently under revision at the Annals of Probability.

Fei Lu
Johns Hopkins University

Statistical Learning and Inverse Problems for Interacting Particle Systems
Systems of interacting particles/agents arise in multiple disciplines, such as particle systems in physics, flocking birds and migrating cells in biology, and opinion dynamics in social science. An essential task in these applications is to learn the rules of interaction from data. We study the nonparametric regression estimator for the pairwise interaction kernels from trajectory data of differential systems, with examples including opinion dynamics, the Lennard-Jones system, and mean-field PDEs. When the system has finite particles, we have a statistical learning problem, and we provide a systematic learning theory addressing the fundamental issues, such as identifiability and mini-max convergence rate for the estimator. When the system has infinite particles, we have an ill-posed inverse problem for PDEs, and we introduce a new regularization method using adaptive RKHSs. Furthermore, learning kernels in operators emerges as a new topic unifying the two settings, and we discuss related open questions.

Hongwei Mei
Texas Tech University

The Minimax Wiener Sequential Testing Problem
Consider the sample path of a one-dimensional diffusion for which the diffusion coefficient is given and where the drift may take on one of two values with an unknown priori distribution. Given an initial state for the observed process, a minimax formulation of the Wiener sequential testing problem (with non-constant signal-to-noise ratio) is considered for detecting the correct drift coefficient as soon as possible and with minimal probabilities of incorrect terminal decisions. The existence of a least favorable (priori) distribution is investigated and its characterization is derived. When the signal-to-noise ratio is a constant particularly, it is shown that the least favorable distribution is half and half.

Athanasios (Sakis) Christ Micheas
University of Missouri

Random Mixture Cox Point Process
We introduce and study a new class of Cox point processes, based on random mixture models of exponential family components for the intensity function of the underlying Poisson process. We investigate theoretical properties of the proposed probability distributions of the point process, including Laplace and other functionals, as well as densities (Radon-Nikodym derivatives) of the probability measure involved. In addition, we provide procedures for parameter estimation using a classical and Bayesian approach. An application of the methodology is presented, where we model the stochastic process of Tornado occurrences in the Midwest.

Aaron Molstad
University of Minnesota

Multiresolution Categorical Regression
In many categorical response regression applications, the response categories admit a multiresolution structure. That is, subsets of the response categories may naturally be combined into coarser response categories. In such applications, practitioners are often interested in estimating the resolution at which a predictor affects the response category probabilities. In this talk, we propose a method for fitting the multinomial logistic regression model in high dimensions that addresses this problem in a unified and data-driven way. Our method allows practitioners to identify which predictors distinguish between coarse categories but not fine categories, which predictors distinguish between fine categories, and which predictors are irrelevant. For model fitting, we propose a scalable algorithm that can be applied when the coarse categories are defined by either overlapping or nonoverlapping sets of fine categories. Statistical properties of our method reveal that it can take advantage of this multiresolution structure in a way existing estimators cannot. We use our method to model cell-type probabilities as a function of a cell's gene expression profile (i.e., cell-type annotation). Our fitted model provides novel biological insights which may be useful for future automated and manual cell-type annotation methodology.

Farzad Sabzikar
Iowa State University

Advanced Perturbation Techniques in Stochastic Optimization for Complex Data Spaces
In this talk, we explore recent advancements in stochastic optimization specifically designed for high-dimensional data. Conventional optimization algorithms often struggle with non-convex landscapes, particularly in overcoming local minima. To tackle these challenges, we introduce heavy-tailed noise models integrated into gradient-based techniques. By employing perturbation strategies and analyzing correlated noise, we enhance both exploration capabilities and convergence efficiency. These developments provide greater robustness in optimizing complex, high-dimensional problems, with wide-ranging implications for machine learning and data science.

Li-Cheng Tsai
University of Utah

Stochastic Heat Flow by Moments
The critical two-dimensional Stochastic Heat Flow (SHF) is the scaling limit of the directed polymers in random environments and the noise-mollified Stochastic Heat Equation (SHE), at the critical dimension of two and near the critical temperature. Caravenna, Sun, and Zygouras (2023) proved that the discrete polymers converge to a universal (model-independent) limit, thereby identifying the limit as the SHF. In this talk, I will demonstrate a different, axiomatic approach to the SHF. We formulate a set of axioms for the SHF, prove the uniqueness in law under these axioms, and also prove the existence by showing that the noise-mollified SHE converges to the SHF under this formulation. The key item in the axioms requires the matching of the first four moments.

Guanyang Wang
Rutgers University

MCMC When You Do Not Want to Evaluate the Target Distribution
In sampling tasks, it is common for target distributions to be known up to a normalizing constant. However, in many situations, evaluating even the unnormalized distribution can be costly or infeasible. This issue arises in scenarios such as sampling from the Bayesian posterior for large datasets and the ‘doubly intractable’ distributions. We provide a way to unify various MCMC algorithms, including several minibatch MCMC algorithms and the exchange algorithm. This framework not only simplifies the theoretical analysis of existing algorithms but also creates new algorithms. Similar frameworks exist in the literature, but they concentrate on different objectives.

Xuan Wu
University of Illinois, Urbana-Champaign

From the KPZ Equation to the Directed Landscape
This talk presents the convergence of the KPZ equation to the directed landscape. We will emphasize on the role played by directed polymers and Gibbsian line ensembles.

George Yin
University of Connecticut

Stochastic Kolmogorov Systems and Applications
In this talk, we present some of our recent work on stochastic Kolmogorov systems. The motivation stems from dealing with important issues of ecological and biological systems. Focusing on environmental noise, we aim to address such fundamental questions: "what are the minimal conditions for long-term persistence of a population, or long-term coexistence of interacting species". [The talk reports some of our joint work with D.H. Nguyen and N.N Nguyen, and also the work with N.T. Dieu, N.H. Du, among others.]

Poster Presentations

Emmanuel Masavo Djegou
Missouri University of Science and Technology

Accelerated Gap Time Models With Fixed Covariates Under A Family Of Effective Age Processes
This work introduces a general class of accelerated gap time models that extend existing accelerated life models to assess the impact of interventions on system failures by shifting time scales. These models have broad applications in reliability engineering test for identifying product weaknesses, as well as in real estate, actuarial science, medicine for predicting depreciation, retirement, and intervention outcomes. The models incorporate a family of effective age processes that captures various aging patterns and fixed covariates that have the potential to accelerate or decelerate the event occurrences. A class of rank-based estimators is constructed, ensuring consistency and asymptotic normality under efficient semiparametric conditions.  Additionally, an Aalen-type nonparametric maximum likelihood estimator is proposed for the baseline intensity function. A simulation studies demonstrate the superior performance of the proposed models compared to existing approaches. A real-world application to the bladder tumor data of Byar (1980) illustrates the practical utility of the models in analyzing recurrent event data.

Dung Quang Le
University of Texas

Rate of Convergence in the Smoluchowski-Kramers Approximation for Mean-field Stochastic Differential Equations
In this research we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski- Kramers approximation) in the L^p -distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes.  This is joint work with Ta Cong Son and Manh Hong Duong.

Jeonghwa Lee
University of North Carolina-Wilmington

Scaling Limit of Dependent Random Walks
Recently, a generalized Bernoulli process 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 generalized Bernoulli processes 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.

Pilhwa Lee
Morgan State University

Debiasing Sinkhorn Divergence in Optimal Transport of Cellular Dynamics
Single-cell RNA-seq analysis helps characterize developmental mechanisms of cellular differentiation, lineage determination, and reprogramming with differential microenvironment conditioning. The underlying dynamics are formulated via dynamic optimal transport (OT) algorithms that calculate cell dynamics over time. We explore the algorithmic biases of OT with entropic regularization by altering transport distance. Sinkhorn divergence debiases these algorithms normally by centering them. This study investigates whether regularized and debiased OT algorithms can characterize cell types within datasets of reprogramming of mouse embryonic fibroblasts and stratifying epidermis morphogenesis.

Mai Nguyen
University of Connecticut

Tempered Bohl-Perron Theorem and Hyperbolicity for Infinite-Dimensional Random Differential Equations
We consider the relationship between the tempered exponential stability and tempered hyperbolicity for linear infinite dimensional random dynamical systems over abstract dynamical systems. Starting with the definitions of tempered exponential stability and tempered hyperbolicity for abstract dynamical systems, we derive necessary and sufficient conditions for tempered exponential stability. Our result establishes the connection between tempered exponential stability and tempered exponentially hyperbolicity. This is a joint work with Prof. N.H.Du and Dr. N.T.T.Nga.

Hongjiang Qian
Auburn University

Moderate Deviation Principles for Stochastic Differential Equations in a Fast Markovian Environment
In this paper, we investigate the moderate deviation principles for a fully coupled two-time-scale stochastic systems. The slow process is governed by stochastic differential equations, while the fast process is a rapidly changing purely jump process on finite state space. The system is fully coupled, meaning the drift and diffusion coefficients of the slow process, as well as the jump distribution of the fast process, are influenced by the the states of both processes. The diffusion component may be degenerate. Our approach is based on the combination of the weak convergence method from [A. Budhiraja, P. Dupuis, and A. Ganguly, Electron. J. Probab. 23 (2018), pp. 1-33; Ann. Probab. 44 (2016), pp. 1723-1775] with the Poisson equation approach for the fast-varying purely jump process.