The Probability seminar is held on Tuesday 4:05pm - 5:05pm in Ayres Hall.

Fall 2025

• Tuesday 11/25: Jeremy Clark. University of Mississippi.

• Tuesday 11/11: Paramahansa Pramanik, University of South Alabama.

• Tuesday 10/14: Dominic Bair, University of Tennessee Knoxville.

• Tuesday 09/30: Barkat Mian, University of Tennessee Knoxville.

Title: On planar Brownian motion singularly tilted through a point potential. Part II.

Abstract: We discuss a special family of two-dimensional diffusions, defined over a finite time interval [0,T]. These diffusions have  transition density functions that are given by the integral kernels of the semigroup corresponding to the two-dimensional Schrödinger operator with a point potential at the origin. Although, in a few ways, our processes of interest are closely related to two-dimensional  Brownian motion,  they have a singular drift pointing in the direction of the origin that is strong enough to enable the possibility of visiting there with positive probability. Our main focus is on  characterizing a local time process at the origin for these diffusions analogous to that for a one-dimensional Brownian motion. This is joint work with Jeremy Clark. 

• Tuesday 09/16: Barkat Mian, University of Tennessee Knoxville.

Title: On planar Brownian motion singularly tilted through a point potential. Part I

Abstract: We discuss a special family of two-dimensional diffusions, defined over a finite time interval [0,T]. These diffusions have  transition density functions that are given by the integral kernels of the semigroup corresponding to the two-dimensional Schrödinger operator with a point potential at the origin. Although, in a few ways, our processes of interest are closely related to two-dimensional  Brownian motion,  they have a singular drift pointing in the direction of the origin that is strong enough to enable the possibility of visiting there with positive probability. Our main focus is on  characterizing a local time process at the origin for these diffusions analogous to that for a one-dimensional Brownian motion. This is joint work with Jeremy Clark. 

• Tuesday 09/09: Hung Nguyen, University of Tennessee Knoxville.

Title: Exponential mixing for the stochastic Kuramoto-Sivashinsky equation on the 1D torus.

Abstract: We discuss the large-time behaviors of the Kuramoto-Sivashinsky equation (KSE) on the 1D torus while being subjected to random perturbation via additive Gaussian noise. It is well-known that under suitable assumptions on the stochastic forcing, the KSE admits a unique statistically steady state. In this work, we make further progress on the topic of ergodicity by addressing the issue of convergence rate toward equilibrium. In particular, we demonstrate that the unique invariant probability measure is exponentially attractive with respect to a suitable Wasserstein distance. This talk is based on a joint work with Peng Gao.

• Tuesday 09/02: Hung Nguyen, University of Tennessee Knoxville.

Title: Ergodicity of the stochastic Kuramoto-Sivashinsky equation. 

Abstract: The deterministic Kuramoto-Sivashinsky equation (KSE) was originally developed in 1970s to model instabilities in reaction-diffusion systems and propagation of laminar flame front. Since then, it has become an important equation known to exhibit chaos and has been investigated through a vast literature. In this talk, we consider the 1D KSE in the presence of random perturbations. Under suitable conditions on the noise terms, we will discuss the well-posedness and the large-time stability, all of which are based on the work of Ferrario, Stoch. Anal. Appl. 26(2): 379-407, 2008. 

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. 

Spring 2024:

• Tuesday 04/30: Daniel McBride, University of Tennessee Knoxville.

Title: Constrained Hamiltonian Monte Carlo for conditional diffusion, Part III.

• Tuesday 04/23: Daniel McBride, University of Tennessee Knoxville.

Title: Constrained Hamiltonian Monte Carlo for conditional diffusion, Part II.

• Tuesday 04/16: Daniel McBride, University of Tennessee Knoxville.

Title: Constrained Hamiltonian Monte Carlo for conditional diffusion, Part I.

Abstract: The goal of this talk is the exposition and exploration of probability theoretical foundations for a variation of constrained path space sampling as laid out by Beskos et al. (Stoch. Process. Their Appl. 121(10):2201–2230, 2011) which avoids the requirement of closed form expressions for a large class of target processes. To motivate the problem in a simpler setting, I will outline some probability theoretical aspects of constrained Hamiltonian Monte Carlo for sampling a class of singular conditional distributions on finite dimensional state spaces arising from equality constraints with smooth zero locus. Then, I will describe some special considerations which must be made in the infinite dimensional path space setting of an analogous class of conditional diffusions.

• Tuesday 04/02: Kihoon Seong, Cornell University.

Title: Phase transition issue of focusing Gibbs measures for Schrödinger-wave systems.

Abstract: We study the phase transition phenomenon of the singular Gibbs measure associated with the Schrödinger-wave systems, initiated by Lebowitz, Rose, and Speer (1988). In the three-dimensional case, this problem turns out to be critical, exhibiting a phase transition according to the size of the coupling constant. In the weakly coupling region, the Gibbs measure can be constructed as a probability measure, which is singular with respect to the Gaussian free field. On the other hand, in the strong coupling case, the Gibbs measure can not be normalized as a probability measure. In particular, the finite-dimensional truncated Gibbs measures have no weak limit, even up to a subsequence.

• Tuesday 03/26: Xiong Wang, Johns Hopkins University.

Title:  Optimal minimax rate of learning interaction kernels.

Abstract: Nonparametric estimation of nonlocal interaction kernels is crucial in various applications involving interacting particle systems. A central question is whether the optimal minimax rate of convergence for this problem aligns with the rate in classical nonparametric regression. Our study confirms this alignment for systems with a finite number of particles. We introduce a tamed least squares estimator (tLSE) that attains the optimal convergence rate for a broad class of exchangeable distributions. The tLSE bridges the smallest eigenvalue of random matrices and Sobolev embedding. The upper minimax rate relies on nonasymptotic estimates for the left tail probability of the smallest eigenvalue of the normal matrix. The lower minimax rate is derived using the Fano-Tsybakov hypothesis testing method. 

• Tuesday 03/19: Jian Song, Shandong University.

Title: On a class of stochastic fractional heat equations.

Abstract: For the fractional heat equation $\frac{\partial}{\partial t} u(t,x) = -(-\Delta)^{\frac{\alpha}{2}}u(t,x)+  u(t,x)\dot W(t,x)$ where  the covariance function of the Gaussian noise $\dot W$ is defined by the heat kernel, we establish Feynman-Kac formulae for both  Stratonovich and Skorohod solutions, along with their respective moments. In particular, we prove that  $d<2+\alpha$ is a sufficient and necessary condition for the equation to have a unique square-integrable mild Skorohod solution.  One motivation lies in the occurrence of this equation in the study of a random walk in  random environment which is generated by a field of independent random walks starting from a Poisson field.  This is joint work with Meng Wang and Wangjun Yuan. 

• Tuesday 02/20: Xia Chen, University of Tennessee, Knoxville.

Title: Hyperbolic Anderson equations with time-independent Gaussian noise: Stratonovich regime - Part II.

• Tuesday 02/13: Xia Chen, University of Tennessee, Knoxville.

Title: Hyperbolic Anderson equations with time-independent Gaussian noise: Stratonovich regime - Part I.

Abstract: In this talk, we shall report some progress made for the hyperbolic Anderson model in Stratonovich regime. Our approach provides new ingridient on representation and computation for Stratanovich moments. The work is based on a collaborative project with Yaozhong Hu.

Fall 2023

• Tuesday 11/07: Christian Houdré, Georgia Institute of Technology.

Title: On Stein's Method for Infinitely Divisible Laws.

Abstract: Stein's method is a well-known methodology to quantify the proximity of probability measures, which has been mainly developed in the Gaussian and Poisson settings. In this talk, I will present some relatively recent unifying results on Stein's method for infinitely divisible laws. This talk is based on joint works with Benjamin Arras. 

• Tuesday 10/24: Yizao Wang, University of Cincinnati.

Title: On the computation of limits of stationary measures of asymmetric simple exclusion processes with open boundary.

Abstract: Recently, several advances have been made on limits of stationary measures of asymmetric simple exclusion processes with open boundary (open ASEP). This talk will present an overview on the method underlying many of these developments. The starting point of the method is a new representation of the probability generating function of stationary measures of open ASEP in terms of the so-called Askey-Wilson Markov processes introduced by Bryc and Wesolowski (2010, 2017). Compared to the well-known Derrida’s matrix ansatz from the 1990s, the advantage of this new representation is that for the Laplace transform of stationary measures of open ASEP it becomes much easier to compute its asymptotics. Computing the limit of Laplace transform then leads to the limit of stationary measures. This conceptually straightforward computation, however, consists of two further delicate steps. One is the computation of the so-called tangent process of Askey-Wilson process, and the other is the derivation of a duality formula of Laplace transforms of certain Markov processes. The talk shall present how the method has been applied and adapted, particularly regarding these two steps, in various setups of open ASEP, and comment on other potential applications. Based on a series of joint works with Wlodek Bryc, Alexey Kuznetsov, Jacek Wesolowski and Zongrui Yang. 

• Tuesday 09/19: Hung Nguyen, University of Tennessee, Knoxville.

Title: Ergodicity of the generalized Langevin equation with singular potentials.

Abstract: We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori-Zwanzig approach, we represent the GLE by a system of SDEs and investigate the large-time stability of the Markov dynamics. Under a general set of conditions on the nonlinearities, we demonstrate that the Markov GLE is always exponentially attractive toward the unique invariant Gibbs measure. Important examples of singular potentials in our results include the Lennard-Jones function and the Coulomb function. This talk is based on a joint work with Manh Hong Duong.

• Tuesday 09/12: Hung Nguyen, University of Tennessee, Knoxville.

Title:  Short Memory Limit for long-time Statistic in a Stochastic Coleman-Gurtin Model of Heat Conduction.

Abstract: We discuss the stochastic Coleman-Gurtin model, a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporate memory effects while subjected to random perturbations via an additive Gaussian noise. In the first part of the talk, we describe the large-time statistically steady states of the equation. Then, in the short memory limit, i.e., as the memory collapses to a Dirac mass, we establish the system's finite time and time asymptotic validity toward the classical reaction-diffusion equation. The arguments require estimates of suitable Wassernstein distances, which must be uniform regarding the memory. This talk is based on a joint work with Nathan Glatt-Holtz and Vincent Martinez.