Probability Archives for Fall 2024 to Spring 2025
Antichain Codes and Anticoncentration
When: Wed, September 27, 2023 - 2:00pmWhere: Kirwan Hall 1310
Speaker: Benjamin Gunby (Rutgers University) - https://sites.google.com/view/benjamingunby/home
Abstract: Let S be a subset of the Boolean cube that is both an antichain and a distance-2r+1 code. How large can S be? I will describe the answer to this question and link it to combinatorial proofs of anticoncentration inequalities for sums of random variables. Joint work with Xiaoyu He, Bhargav Narayanan, and Sam Spiro.
A new front in multidimensional branching Brownian motion
When: Wed, October 18, 2023 - 2:00pmWhere: Kirwan Hall 1310
Speaker: Yujin Kim (NYU Courant) - https://yujinhkim.github.io/
Abstract: Branching Brownian motion (BBM) has gained lots of attention in recent years in part due to its significance to the universality class of log-correlated fields and their extremal landscapes. Recently, J. Berestycki, K., Lubetzky, Mallein and Zeitouni obtained a description of the limiting point process of the extreme values and locations of BBM in dimensions 2 and higher; however, certain details of the extremal landscape were washed out in that limit. In this talk, we describe a more precise limiting point process that recovers those details and gives rise to a new object in the study of BBM, what we call the extremal front. We then obtain a scaling limit, of exponent 3/2, for the extremal front. Joint work with Ofer Zeitouni.
Parabolic equations and diffusion processes with degeneration: boundary problems, metastability, and homogenization
When: Thu, November 2, 2023 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Leonid Koralov (UMD) -
Abstract: This is a joint talk with the PDE seminar.
Stated in PDE terms, the problems concern the asymptotic behavior of solutions to
parabolic equations whose coefficients degenerate at the boundary of a domain.
The operator may be regularized by adding a small diffusion term. Metastability effects arise in this case: the asymptotics of solutions, as the size of the perturbation tends to zero, depends on the time scale. Initial-boundary value problems with both the Dirichlet and the Neumann boundary conditions are considered. We also consider periodic homogenization for operators with degeneration. The talk is based on joint work with M. Freidlin.
Large deviations of the KPZ equation
When: Wed, November 15, 2023 - 2:00pmWhere: Math 1310
Speaker: Yier Lin (University of Chicago) - https://yierlin.me/
Abstract: The KPZ equation is a stochastic PDE that plays a central role in a class of random growth phenomena. In this talk, we will explore the large deviations for the KPZ equation through the lens of the variational principle. We will explain how to extract various limits of the most probable shape of the KPZ equation from the variational formula. Our method combines the ideas from probability and PDE. This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.
Matching upper and lower moment bounds for a large class of stochastic PDEs driven by general space-time Gaussian noises
When: Thu, November 16, 2023 - 2:30pmWhere: Math 1313
Speaker: Yaozhong Hu (University of Alberta at Edmonton) - https://sites.ualberta.ca/~yaozhong/
Abstract: I will present some results obtained with Dr. Xiong Wang on the matching upper and lower moment bounds of the solution for a large class of stochastic partial differential equations driven by a general Gaussian noises, giving a rather complete answer to the open problem of the matching lower moment bounds for the stochastic wave equations for general Gaussian noises. Two new general conditions are introduced for the Green's function of the equation to assure this intermittency property: small ball nondegeneracy and bounded Hardy-Littlewood-Sobolev total mass, which are satisfied by a large class of stochastic PDEs, including stochastic heat equations, stochastic wave equations, stochastic heat equations with fractional Laplacians, and stochastic partial differential equations with fractional derivatives both in time and in space. The main technique to obtain the lower moment bounds is to develop a Feynman diagram formula for the moments of the solution, to find the manageable main terms, and to carefully analyse these terms of sophisticated multiple integrals by exploring the above two properties.
A journey through greediness and dependences
When: Wed, November 29, 2023 - 2:00pmWhere: Math 1310
Speaker: Andrea Ottolini (University of Washington) - https://faculty.washington.edu/ottolini/
Abstract:
This talk is really three talks. In the first part, our main focus is an algorithm used to approximately solve an optimal transport problem that often arises in economic applications: how do we minimize the cost of producers and consumers, when the cost is a concave function of the distance? In the second part, we analyze how feedback can be used to improve the performance of certain card-guessing games, closely related to well-known statistical problems such as bias-detection in clinical trials and hypothesis testing. In the third part, motivated by a novel notion of curvature on graphs, we provide some close-to-exact formulas for the hitting times in Erdös-Rényi random graphs. These stories are held together by a common theme: sometimes, greediness and dependencies are not that harmful. This is based on joint works with J. He, S. Steinerberger and R. Tripathi.
RIT on Rough Paths
When: Mon, February 5, 2024 - 2:00pmWhere: Kirwan Hall 2300
Speaker: Sandra Cerrai (UMD) -
Long-time behaviour for granular media equation
When: Wed, February 7, 2024 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Julian Tugaut (Jean Monnet University) - https://tugaut.perso.math.cnrs.fr/home.html
Abstract: Starting from the microscopical viewpoint with particles system, we briefly derive the associated Partial Differential Equation that is denoted as granular media equation. The aim is to find out the limiting steady state or at least to find suitable conditions ensuring that the limiting steady state is a given invariant probability measure. We will recall that such measure is unique under convexity assumption and that the convergence holds. Then, we will discuss the nonconvex case, where there is convergence albeit non-uniqueness of the invariant probability measures may occur. In a second time, we will give the main theorem that is a stability type theorem: if the initial measure is sufficiently close to a given stationary measure then this measure is the limiting steady state. Finally, we will give the proof of the theorem. If time allows us, we will discuss on possible generalizations. This talk is based on a work published in Kinetic and Related Models.
Convergence of Rain Process Models
When: Wed, February 21, 2024 - 2:00pmWhere: Kirwan Hall 1310
Speaker: Scott Hottovy (Department of mathematics, US Naval Academy) - https://www.usna.edu/Users/math/hottovy/index.php
Abstract: Representing rain in global climate models continues to be a challenge. Currently, models generally rain too often and too little. Additionally, the models have trouble capturing variability in rain data. One possible solution to increasing variability in a model is to use a stochastic process. A variety of stochastic models have been used to describe time series of precipitation or rainfall. Since many of these stochastic models are simplistic, it is desirable to develop connections between the stochastic models and the underlying physics of rain. In this talk, I will describe simple models of rain in a single column model as a stochastic differential equation (SDE) with a switch. The inclusion of this switch leads to a model with hysteresis. I will show how these models are connected by presenting formal derivations and theorems on convergence of SDEs and their Kolmogorov Equations.
A Gaussian convexity for logarithmic moment generating function
When: Tue, March 26, 2024 - 4:00pmWhere: MATH 1308
Speaker: Wei-Kuo Chen (UMN) - https://sites.google.com/view/wkchen/home
Abstract: Convex functions of Gaussian vectors are prominent objectives in many fields of mathematical studies. In this talk, I will establish a new convexity for the logarithmic moment generating function for this object and draw two consequences. The first leads to the Paouris-Valettas small deviation inequality that arises from the study of convex geometry. The second provides a quantitative bound for the Dotsenko-Franz-Mezard conjecture in the Sherrington-Kirkpatrick mean-field spin glass model, which states that the logarithmic anneal partition function of negative replica is asymptotically equal to the free energy.
Equations on Wasserstein space and applications
When: Wed, May 1, 2024 - 2:00pmWhere: Math 1308
Speaker: Ben Seeger (UT Austin) - https://web.ma.utexas.edu/users/bs42827/
Abstract: The purpose of this talk is to give an overview of recent work involving differential equations posed on spaces of probability measures and their use in analyzing mean field limits of controlled multi-agent systems, which arise in applications coming from macroeconomics, social behavior, and telecommunications. Justifying this continuum description is often nontrivial and is sensitive to the type of stochastic noise influencing the population. We will describe settings for which the convergence to mean field stochastic control problems can be resolved through the analysis of the well-posedness for a certain Hamilton-Jacobi-Bellman equation posed on Wasserstein spaces, and how this well-posedness allows for new convergence results for more general problems, for example, zero-sum stochastic differential games of mean-field type.