Spring 2025
Tuesay 05/06: Xiaoqin Guo, University of Cincinati.
Title: On the rate of convergence of the martingale central limit theorem in Wasserstein distances.
Abstract: We consider the rate of convergence of the central limit theorem (CLT) for martingales. For martingales with a wide range of integrability, we will quantify the rate of convergence of the CLT via Wasserstein distances of order 𝑟, 1 ≤ 𝑟 ≤ 3. Our bounds are in terms of Lyapunov’s coefficients and the $L^{r/2}$ fluctuation of the total conditional variances. Our Wasserstein-1 bound is optimal up to a multiplicative constant.
Tuesay 04/29: Ray (Shuyang) Bai , University of Georgia.
Title: An Unusual Example of Extremal Clustering.
Abstract: Extreme value theory for stationary processes has been well developed over the past decades. Central to the theory is the notion of the extremal index, which quantifies the strength of clustering of extreme values due to dependence and appears as a correcting factor in the limit distribution. Classical theory provides a predictive formula for the extremal index, known as the candidate extremal index, which is typically equal to the extremal index under mild regularity conditions. In this talk, we introduce a double series model constructed via intersections of independent discrete renewal processes with infinite mean. This model exhibits an unusual phenomenon: the extremal index and the candidate extremal index differ. We further discuss several other surprising consequences that arise from this discrepancy. The talk is based on joint works with Rafal Kulik and Yizao Wang.
Tuesay 04/22: Jan Rosinski , University of Tennessee Knoxville.
Title: Continuous time random walks, Riemann zeta function, and related probability distributions.
Abstract: This talk is a survey of some recent results on the above topics, which will be presented together.
Friday 03/28: Renming Song, UIUC (Colloquium).
Title: Heat kernel estimates for (fractional) Laplacians with supercritical killings.
Abstract: In this talk, I will present some recent results on sharp two-sided estimates on the heat kernels of ( fractional) Laplacians with supercritical killing potentials, that is, heat kernels of operators of the form
$$ -(-\Delta)^{\alpha/2}-\kappa (x) $$
where $\alpha\in (0, 2]$ and $\kappa$ belongs to a class of positive supercritical potentials including $\kappa(x)=c|x|^{-\beta}$ with $\beta>\alpha$. This talk is based on a joint paper with Soobin Cho, and a joint paper with Soobin Cho and Panki Kim.
This talk is aimed at a general audience and I will start with some basic materials.
Tuesday 03/11: Xiao Shen, University of Utah.
Title: Random growth models and the KPZ Universality Class.
Abstract: Many two-dimensional random growth models, including first-passage and last-passage percolation, are conjectured to fall within the Kardar–Parisi–Zhang (KPZ) universality class under mild assumptions on the underlying noise. In recent years, researchers have focused on a subset of exactly solvable models, where these conjectures can be rigorously verified. This talk discusses a specific line of research that focuses on advancing the understanding of both exactly solvable and non-solvable KPZ models using general probabilistic methods, such as percolation and coupling.
Tuesday 03/04: Hung Nguyen, University of Tennessee Knoxville.
Title: Ergodicity of the stochastic nonlinear Schrodinger equation. Part II.
Abstract: We discuss the long time behaviors of the nonlinear Schrodinger equation with damping while being subjected to a random perturbation via Gaussian additive noise. It is well-known that, under the impact of sufficiently strong damping, the dynamics possesses unique ergodicity, albeit lacking a convergence speed. In this talk, we address the issue of mixing property for the large damping scenario and establish that the solutions are attracted toward equilibrium with polynomial rates of any order. While existing results in literature typically employ a Foias-Prodi typed argument, our approach relies on a coupling strategy making use of pathwise Strichartz estimates specifically derived for the equation. This talk is based on a joint work with Kihoon Seong.
Tuesday 02/25: Hung Nguyen, University of Tennessee Knoxville.
Title: Ergodicity of the stochastic nonlinear Schrodinger equation. Part I.
Abstract: We consider the nonlinear Schrodinger equation in the presence of a damping force while being subjected to a random perturbation via Gaussian additive noise. Under different assumptions on the nonlinearity, the noise structure and the spatial dimensions, we will walk through the global well-posedness, the existence and the uniqueness of the statistically steady states. The presented results are respectively based on the work of de Bouard-Debussche 2003, Ekren-Kukavica-Ziane 2017 and Brzezniak-Ferrario-Zanella 2023.
Tuesday 02/04: Cheng Ouyang, University of Illinois Chicago.
Title: Global geometry and parabolic Anderson model on compact manifolds.
Abstract: We introduce a family of intrinsic Gaussian noises on compact manifolds that we call “colored noise” on manifolds. With this noise, we study the parabolic Anderson model (PAM) on compact manifolds. Under some curvature conditions, we show the well-posedness of the PAM and provide some preliminary (but sharp) bounds on the second moment of the solution. It is interesting to see that global geometry enters the play in the well-posedness of the equation.
Fall 2024:
Tuesday 11/12: Benjamin Cooper Boniece, Drexel University.
Title: An iterative approach to volatility estimation
Abstract: The quadratic variation of a semimartingale plays an important role in a variety of applications, particularly so in financial econometrics, where it is closely linked to volatility. In the past two decades, many methods have been put forth that aim to provide jump-robust estimates of volatility by separating quadratic variation into its continuous and discontinuous parts.
However, in spite of the favorable asymptotic statistical properties of these approaches, they face a "tuning problem": their use in practice requires entirely heuristic selection of tuning parameters which can greatly impact their estimation performance.
In this talk, I will discuss some recent work concerning an "automated" iterative approach that circumvents the tuning problem.
Tuesday 10/15: Quyuan Lin, Clemson University.
Title: On the stochastic hydrostatic Euler equations.
Abstract: The hydrostatic Euler equations, also known as the inviscid primitive equations, are utilized to describe the motion of inviscid fluid flow in a thin domain, such as the ocean and atmosphere on a planetary scale. It is well-known that this model is ill-posed in Sobolev spaces and Gevrey class of order greater than one, and its analytic solutions can form singularity in finite time.
In this talk, I will present two recent results on the stochastic version of this model. Firstly, with general multiplicative noise, I will demonstrate how the local Rayleigh condition can be used to address the issue of ill-posedness in Sobolev spaces, leading to the establishment of the existence and uniqueness of pathwise solutions. Secondly, I will discuss that some specific random noises can restore the local well-posedness and prevent finite-time blowups.
Tuesday 10/01: Jorge M Ramirez, Oak Ridge National Laboratory.
Title: Stochastic predator-prey models: noise-induced extinctions and invariance.
Abstract: We study the properties of the Gause model with noise as an illustrative example of the effect that uncertainty can play in non-linear dynamical systems. These equations model the dynamics of a predator-prey system where the predation rate is given by a functional response of the prey, and the prey obeys logistic growth. The noise models the uncertainty in the equation’s parameters and is included as an additive but non-linear Gaussian process. We show conditions for existence and boundedness of the solutions for different types of Holling functional response models. We also show the possibility of noise-induced extinction events in situations where the deterministic dynamics allow for coexistence. Finally, we give conditions on the noise and model parameters for the existence of an invariant distribution.
Thursday 09/19: Xia Chen, University of Tennessee Knoxville. (jointly with PDE seminar)
Title: A quenched spatial law for KPZ and parabolic Anderson equation, part II.
Thursday 09/12: Xia Chen, University of Tennessee Knoxville. (jointly with PDE seminar)
Title: A quenched spatial law for KPZ and parabolic Anderson equation, part I.
Abstract: The KPZ equation was introduced by the physicists Kardar M., Parisi, G. and Zhang, Y. C. (1986) for modeling of growing interface. Its challenge to mathematical community leads to a tsunami in the area of stochastic partial differential equations (SPDE). In this talk, we establish the quenched law for the spatial asymptotics of one-dimensional KPZ equation. Under the Hopt-Cole transform, the problem is reduced to the setting of a linear SPDE known as parabolic Anderson equation.