2026 Blue Ridge Probability Day

Event date: Friday, March 13, 2026

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Spring in Blue Ridge Mountains

The second Blue Ridge Probability Day will occur on Friday, March 13, 2026. Location TBA. The speakers are

Schedule

10:30 - 11:30Mark Meckes
11:30 - 1:30Lunch
1:30 - 2:30Eva Loeser
2:30 - 3:00Coffee/tea
3:00 - 4:00Yu Gu
4:00 - 4:30Coffee/tea
4:30 - 5:30David P. Herzog
6:30Dinner

Yu Gu (University of Maryland) - Invariant measure for open KPZ

Consider a random interface growing above the interval \([0,1]\), one may ask: if the slopes at the two boundary points are prescribed, what does the interface look like? This question reduces to understanding the equilibrium state of the open KPZ equation. In recent work of Corwin–Knizel, Barraquand–Le Doussal, and Bryc–Kuznetsov–Wang–Wesołowski, this equilibrium state is described explicitly and identified as a resampled Brownian motion.

In this talk, I will present recent joint work with Alex Dunlap and Tommaso Rosati in which we provide a stochastic-analytic derivation of this equilibrium state. Our approach relies on tools such as change of measure, time reversal, Itô’s formula, and the theory of regularity structures.

David P. Herzog (Iowa State University) - Quantitative positivity of transition densities for random perturbations of Hamiltonian systems

I will present recent results on a class of diffusion processes arising from random perturbations of conservative Hamiltonian systems. Under a set of abstract hypotheses—including basic structural assumptions on the Hamiltonian, a weak Lyapunov structure, and a quantitative notion of smoothing—we prove that transition densities satisfy a sharp, uniform pointwise lower bound over Hamiltonian sublevel sets in the small noise limit. By applying our general theorem, we obtain quantitative minorization estimates for a variety of models including Langevin dynamics, chains of oscillators coupled to heat baths at different temperatures, and finite-dimensional fluid models such as stochastically forced Galerkin truncations of the Navier-Stokes equations and the Lorenz ‘96 system. As a corollary, assuming a stronger Lyapunov structure, our main result yields a sharp exponential rate of convergence to equilibrium in a weighted total variation norm. A central feature of our approach is that it does not require knowledge of the explicit form of the invariant measure, nor even its existence, and hence is broadly applicable to deduce minorization for physically relevant systems where invariant measures are inaccessible.

Eva Loeser (UNC Chapel Hill) - Invariance Principle for d-Dimensional Markov Processes on a Domain

Many interacting particle systems that arise in mathematical physics, integrable probability, and queueing theory exhibit complex interactions when particles are near each other. Such a system of \(d\) particles traversing the real line can be represented as a stochastic process in \(\mathbb{R}^d\), \((X_1,\ldots,X_d)\), where \(X_i\) is the position of the \(i\)th largest particle. In this situation, we have a process that takes values on the Weyl chamber, \(S = \{ x \in \mathbb{R}^d : x_1 \le x_2 \le \cdots \le x_d \}\), where the interesting interaction occurs near the boundary of \(S\) (when at least two particles are near each other). The following question then arises: how can we take the diffusion limit of a Markov process in \(\mathbb{R}^d\) that is constrained to a certain region where, in the microscopic dynamics, there is a proliferation of different boundary behaviors? To answer this question, we explore a generalization of the classical invariance principles for semimartingale reflected Brownian motions (SRBM) explored by Williams, Dai, Reiman, Kang, and others to situations in which the number of boundary behaviors can be much larger than the number of faces. We will use this theorem to obtain clean diffusion limits for examples from mathematical physics, integrable probability, and queueing theory.

Mark Meckes (Case Western Reserve University) - Entropy, diversity, and complexity in metric spaces and beyond

I will discuss a family of emerging, geometrically aware generalizations of Rényi entropies and the theory and myriad connections to other fields growing around them. The story will involve excursions into category theory, theoretical ecology, information theory, convex geometry, probability theory, and more.

Last updated: Monday, January 12, 2026