Spring 2026
Tuesday 04/28: Zhengye Zhou, University of Southern California.
Title: Uniform convergence of Pfaffian Point Process.
Abstract: Recent years have seen significant progress in understanding fluctuations in half-space models within the KPZ universality class. In this talk I will discuss a Pfaffian Schur process, which is a measure on a sequence of partitions, which was introduced by Borodin and Rains as a Pfaffian analog of the determinantal Schur processes introduced by Okounkov and Reshetikhin. The model we investigate arises in a half-quadrant last passage percolation model, which has i.i.d. geometric weights with parameter $\alpha \in (0,1)$ off of the main diagonal and with parameter $c \in (0, \alpha^{-1})$ on the main diagonal. We show that when $c$ is subcritical the line ensembles formed by the parts of our random partitions converge uniformly over compact sets to the Airy line ensemble; when $c$ is critically scaled with the size of the system the ensemble instead converges to the Airy wanderer line ensemble with a single parameter; and when $c$ is supercritical we will see both Gaussian and Airy line ensemble in the limit.
Tuesday 04/07: Ryoji Takano, Tokyo Metropolitan University.
Title: A semigroup approach to the reconstruction theorem and its applications .
Abstract: In classical stochastic analysis, constructing solutions to singular stochastic partial differential equations is a highly nontrivial task. Hairer’s theory of regularity structures made it possible to construct local-in-time solutions under appropriate assumptions, and the reconstruction theorem is one of the key analytic tools in this framework. In this talk, we extend the semigroup approach developed by Otto–Weber (2019) and Hoshino (2023) to provide an alternative proof of the reconstruction theorem. As an application, we construct a local-in-time solution to the two-dimensional parabolic Anderson model with a non-translation-invariant differential operator. The talk will begin with a brief introduction to the regularity structure approach to singular SPDEs and a discussion of how our method differs from previous ones. We will then present our main result, focusing on the reconstruction theorem. This talk is based on joint work with Masato Hoshino (Institute of Science Tokyo).
Tuesday 03/24: Alex Dunlap, Duke University.
Title: Stationary measure for the open KPZ equation: an analytic viewpoint.
Abstract: I will describe an analytic approach to proving the invariance of the stationary measure for the inhomogeneous open KPZ equation, given the knowledge of the (Brownian) stationary measure for the problem with homogeneous boundary conditions. In place of tools from integrable probability, we use a time-reversal argument and Itô’s formula to reduce the problem to a statement about the behavior of the KPZ nonlinearity at the boundary. We then use the theory of regularity structures to prove this statement. Joint work with Yu Gu and Tommaso Rosati.
Tuesday 03/17: Soobin Cho, University of Illinois Urbana-Champaign. (Cancelled due to weather)
Tuesday 03/03: Xia Chen, University of Tennessee Knoxville.
Title: Hyperbolic Anderson models and Brownian motions.
Abstract: This talk addresses a recent discovery that links the hyperbolic equations driven by Gaussian noises to Brownian motions. A part of the talk is based on a recently published paper (Ann. Probab. 2025) joint with Yaozhong Hu, where the Gaussian norse is independent of time. Another part is an on-going project where the Gaussian noise has a fractional time component.
Tuesday 02/24: Jan Rosinski, University of Tennessee Knoxville.
Title: Markov Processes via generalized convolutions. Part II.
Tuesday 02/17: Jan Rosinski, University of Tennessee Knoxville.
Title: Markov Processes via generalized convolutions. Part I.
Abstract: Generalized convolutions are binary, symmetric, and associative operations on probability measures on R_+. Such convolutions were first introduced by J.F.C. Kingman in the seminal work on random walks with spherical symmetry. Kingman’s convolution facilitates the study of Bessel processes. This is a joint work with Prof. Jolanta Misiewicz of Warsaw University of Technology.
Fall 2025
Tuesday 11/25: Jeremy Clark. University of Mississippi.
Title: Continuum polymer measures corresponding to the critical 2d stochastic heat flow.
Abstract: The critical two-dimensional stochastic heat flow (2d SHF) is a two-parameter process of random Borel measures on R^4 derived in a breakthrough article by Caravenna, Sun, and Zygouras as a universal distributional limit of point-to-point partition functions for (1+2)-dimensional models of a directed polymer in a random environment within a critical weak-coupling scaling regime. Recently, Li-Cheng Tsai has provided a list of properties uniquely characterizing the law of the 2d SHF, and he proved that it arises as a distributional limit of solutions to a mollified version of the two-dimensional stochastic heat equation. I will discuss continuum polymer measures associated with the 2d SHF.
Tuesday 11/18: Dominic Bair, University of Tennessee Knoxville.
Title: First-Passage Percolation: Geodesics and Coarse Geometry. Part II.
Abstract: First-passage percolation (FPP) is a classical random growth model introduced to study random metrics, most notably on the integer lattice. Assigning independent random passage times to edges defines a random distance via minimal total passage time between vertices. We study the existence and behavior of infinite and bi-infinite geodesics in this random metric space. Furthermore, we analyze how the random geometry induced by FPP influences coarse geometric properties such as hyperbolicity and Morseness, highlighting the interplay between probabilistic and geometric phenomena.
Tuesday 11/11: Paramahansa Pramanik, University of South Alabama.
Title: Construction of an Optimal Strategy: An Analytic Insight Through Path Integral Control Driven by a McKean-Vlasov Opinion Dynamics.
Abstract: We have constructed a closed-form optimal strategy within a social network using stochastic McKean-Vlasov dynamics. Each agent independently minimizes their dynamic cost functional, driven by stochastic differential opinion dynamics. These dynamics reflect agents' opinion differences from others and their past opinions, with random influences and stubbornness adding to the volatility. To gain an analytic insight into the optimal feedback opinion, we employed a Feynman-type path integral approach with an appropriate integrating factor, marking a novel methodology in this field. Additionally, we utilized a variant of the Friedkin-Johnsen-type opinion dynamics to derive a closed-form optimal strategy for an agent and conducted a comparative analysis.
Tuesday 10/21: Dominic Bair, University of Tennessee Knoxville.
Title: First-Passage Percolation: Geodesics and Coarse Geometry. Part I.
Abstract: First-passage percolation (FPP) is a classical random growth model introduced to study random metrics, most notably on the integer lattice. Assigning independent random passage times to edges defines a random distance via minimal total passage time between vertices. We study the existence and behavior of infinite and bi-infinite geodesics in this random metric space. Furthermore, we analyze how the random geometry induced by FPP influences coarse geometric properties such as hyperbolicity and Morseness, highlighting the interplay between probabilistic and geometric phenomena.
Tuesday 10/14: Dominic Bair, University of Tennessee Knoxville.
Title: Introduction to first-passage percolation and subadditive processes.
Abstract: First passage percolation (FPP) is a classical random growth model introduced as a way to study random metrics on (classically) the lattice. Assigning independent random passage times to edges, one obtains a random distance by minimizing the total time between vertices. This simple construction gives rise to deep geometric and probabilistic questions: does the scaled growth converge to a deterministic shape? how regular is the limit? and what can be said about the geodesics in the space? In this talk, I will introduce the model, explain the subadditive framework underlying the shape theorem, and discuss some key results and open problems concerning geodesics, limit theorems, and the geometry of random metrics.
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.