Talk given at SEPC 2021 in honor of Elizabeth Meckes

We consider an ensemble of N interacting particles modeled by a system of N stochastic differential equations (SDEs). The coefficients of the SDEs are taken to be such that as N approaches infinity, the system undergoes Kac’s propagation of chaos, and is well-approximated by the solution to a McKean-Vlasov Equation. Rare but possible deviations of the behavior of the particles from this limit may reflect a catastrophe, and computing the probability of such rare events is of high interest in many applications. In this talk, we design an importance sampling scheme which allows us to numerically compute statistics related to these rare events with high accuracy and efficiency for any N. Standard Monte Carlo methods behave exponentially poorly as N increases for such problems. Our scheme is based on subsolutions of a Hamilton-Jacobi-Bellman (HJB) Equation on Wasserstein Space which arises in the theory of mean-field control. This HJB Equation is seen to be connected to the large deviations rate function for the empirical measure on the ensemble of particles. We identify conditions under which our scheme is provably asymptotically optimal in N in the sense of log-efficiency. We also provide evidence, both analytical and numerical, that with sufficient regularity of the solution to the HJB Equation, our scheme can have vanishingly small relative error as N increases.

We will explore the role of demographic stochasticity in triggering extinction events in models of large finite populations. While prior works have focused on large fluctuations from quasi-stationary distributions, we instead consider extinction events occurring before entering a metastable state. Since such extinction events require only slight deviations from the mean-field trajectories, we can derive the approximating extinction probability PDE with a modified Robin-type boundary condition. We then investigate the utility of this approximation by comparing to the Lotka-Volterra model as well as the Lotka-Volterra model with logistic growth.

This talk is an overview of my thesis work, which consists of 3 projects exploring the effect of multiscale structure on a class of interacting particle systems called weakly interacting diffusions. In the absence of multiscale structure, we have a collection of N particles, with the dynamics of each being described by the solution to a stochastic differential equation (SDE) whose coefficients depend on that particle's state and the empirical measure of the full particle configuration. It is well known in this setting that as N approaches infinity, the particle system undergoes the ``propagation of chaos,'' and its corresponding sequence of empirical measures converges to the law of the solution to an associated McKean-Vlasov SDE. Meanwhile, in our multiscale setting, the coefficients of the SDEs may also depend on a process evolving on a timescale of order 1/\epsilon faster than the particles. As \epsilon approaches 0, the effect of the fast process on the particles' dynamics becomes deterministic via stochastic homogenization. We study the interplay between homogenization and the propagation of chaos via establishing large deviations and moderate deviations results for the multiscale particles' empirical measure in the combined limit as N approaches infinity and \epsilon approaches 0. Along the way, we derive rates of homogenization for slow-fast McKean-Vlasov SDEs.

In the late 20th century, statistical physicists introduced a chemical reaction model called ballistic annihilation. In it, particles are placed randomly throughout the real line and then proceed to move at independently sampled velocities. Collisions result in mutual annihilation. Many results were inferred by physicists, but it wasn’t until recently that mathematicians joined in. I will describe my trajectory through this model. Expect tantalizing open questions.

This will be the last in his sequence of an introductory lecture on Hypocoercivity for Langevin dynamics. For those who have not attended the previous lectures and are familiar with Langevin dynamics, the talk should be accessible. We will continue our discussion on convergence to equilibrium for second-order Langevin dynamics using the Poincare approach. We'll recap convergence in H^1(\mu) and then we'll talk about the direct L^2(\mu) method of Dolbeault, Mouhot, and Schmeiser, also called the DMS approach.

An emerging way to protect privacy is to replace true data by synthetic data. Medical records of artificial patients, for example, could retain meaningful statistical information while preserving privacy of the true patients. But what is synthetic data, and what is privacy? How do we define these concepts mathematically? Is it possible to make synthetic data that is both useful and private? I will tie these questions to a simple-looking problem in probability theory: how much information about a random vector X is lost when we take conditional expectation of X with respect to some sigma-algebra? This talk is based on a series of papers with March Boedihardjo and Thomas Strohmer.

Pratima Hebbar, Probability Seminar on October 21, 2021

David Aldous, Probability Seminar Sept 30, 2021 TITLE: Can one prove existence of an infectiousness threshold (for a pandemic) in very general models of disease spread? ABSTRACT: Intuitively, in any kind of disease transmission model with an infectiousness parameter, there should exist a critical value of the parameter separating a very likely from a very unlikely resulting pandemic. But even formulating a general conjecture is challenging. In the most simplistic model (SI) of transmission, one can prove this for an essentially arbitrary large weighted contact network. The proof for SI depends on a simple lemma concerning hitting times for increasing set-valued Markov processes. Can one extend to SIR or SIS models over similarly general networks, where the lemma is no longer applicable?

SEPC 2021 in honor of Elizabeth Meckes. Slides from the talks and more information are available <a href="https://services.math.duke.edu/~rtd/SEPC2021/SEPC2021.html">here</a>

SEPC 2021 in honor of Elizabeth Meckes. Slides from the talks and more information are available <a href="https://services.math.duke.edu/~rtd/SEPC2021/SEPC2021.html">at this link (here).</a>

Talk given at SEPC 2021 in honor of Elizabeth Meckes

Talk given at SEPC 2021 in honor of Elizabeth Meckes

Talk given at SEPC 2021 in honor of Elizabeth Meckes

Description of some work with Elizabeth Meckes at SEPC 2021

Quasi-Stationary Distributions (QSDs) describe the long-time behaviour of killed Markov processes. The Fleming-Viot particle system provides a particle representation for the QSD of a Markov process killed upon contact with the boundary of its domain. Whereas previous work has dealt with killed Markov processes, we consider killed McKean-Vlasov processes. We show that the Fleming-Viot particle system with McKean-Vlasov dynamics provides a particle representation for the corresponding QSDs. Joint work with James Nolen.

A key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology, which says in its simplest form that two species competing for one resource cannot coexist, and show how the random switching can facilitate coexistence.

There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term. These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge, Freeman, and Pennington to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et al there were two nontrivial stationary distributions.

We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d \ge 3$. Combining this result with properties of the PDE and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first two applications confirm conjectures of Cox and Perkins and Ohtsuki et al.

I will present a tutorial on the mathematical models utilized in molecular biology. I will begin with an introduction to the usual stochastic and deterministic models, and then introduce terminology and results from chemical reaction network theory. I will end by presenting the “deficiency zero” theorem in both the deterministic and stochastic settings.

The graph Laplacian has been of interest in statistics, machine learning, and theoretical computer science in areas from manifold learning to analysis of Markov chains. A common uses of the graph Laplacian has been in spectral clustering and dimension reduction. A theoretical motivation for why spectral clustering works is the Cheeger inequality which relates the eigenvalues of the graph Laplacian to how disconnected the graph is, Betti zero for topology. We ask how the Cheeger inequality extends to higher-order Laplacians, operators on simplicial complexes, and what clustering means for these higher-order operators. This is joint work with John Steenbergen Related to the graph Laplacian is the idea of random walks on graphs. We will define a random walk on simplicial complexes with a stationary distribution that is related to the k-dimensional Laplacian. The stationary distribution reveals (co)homology of the geometry of the random walk. We apply this random walk to the problem of semi-supervised learning, given some labeled observations and many unlabeled observations how does one propagate the labels.