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>

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.

Talk given at SEPC 2021 in honor of Elizabeth Meckes

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.

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.

Description of some work with Elizabeth Meckes at SEPC 2021

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.

Talk given at SEPC 2021 in honor of Elizabeth Meckes

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>

Talk given at SEPC 2021 in honor of Elizabeth Meckes

Talk given at SEPC 2021 in honor of Elizabeth Meckes

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?

Random multiplicative functions are probabilistic models for multiplicative arithmetic functions, such as Dirichlet characters or the Liouville function. In this talk, I will first quickly give an overview of the area, and then focus on some of the recent works on proving central limit theorems, connections to additive combinatorics, as well as some other deterministic applications. Part of the talk is based on joint work with Soundararajan, with Harper and Soundararajan (in progress) and with Angelo and Soundararajan (in progress).

In r-neighbor bootstrap percolation, the vertices of Z^d are initially occupied independently with probability p and empty otherwise. Occupied vertices remain occupied forever, and empty vertices iteratively become occupied when they have at least r occupied neighbors. It is a classic result of van Enter (r=d=2) and Schonmann (d>2 and r between 2 and d) that every vertex in Z^d eventually becomes occupied for any initial density p>0. In the polluted bootstrap percolation model, vertices of Z^d are initially closed with probability q, occupied with probability p and empty otherwise. The r-neighbor bootstrap rule is the same, but now closed vertices act as obstacles, and remain closed forever. This model was introduced 20 years ago by Gravner and McDonald, who studied the case d=r=2 and proved a phase transition exists for this model as p and q tend to 0. We prove a similar phase transition occurs when d=r=3, and we identify the polynomial scaling between p and q at which this transition occurs for the modified bootstrap percolation model. For one direction, our proof relies on duality methods in Lipschitz percolation to find a blocking structure that prevents occupation of the origin. The other direction follows from a rescaling argument, and the recent results of Holroyd and Gravner for d>r=2. This is joint work with Holroyd and Gravner.

For neutral branching models of two types of populations there are three universality classes of behavior: independent branching, (one-sided) catalytic branching and mutually catalytic branching. Loss of independence in the two latter models generates many new features in the way that the populations evolve. In this talk I will focus on describing the genealogy of a catalytic branching diffusion. This is the many individual fast branching limit of an interacting branching particle model involving two populations, in which one population, the "catalyst", evolves autonomously according to a Galton-Watson process while the other population, the "reactant", evolves according to a branching dynamics that is dependent on the number of catalyst particles. We show that the sequence of suitably rescaled family forests for the catalyst and reactant populations converge in Gromov-Hausdorff topology to limiting real forests. We characterize their distribution via a reflecting diffusion and a collection of point-processes. We compare geometric properties and statistics of the catalytic branching forests with those of the "classical" (independent branching) forest. This is joint work with Andreas Greven and Anita Winter.

The talk will contain a review of several recent results on the analytic continuation of the random analytic functions. We will start from the classical theorem on the random Taylor series (going to Borel’ s school), but the main subject will be the random zeta – function (which was introduced implicitly by Cramer) and its generalizations. We will show that “true primes are not truly random “, since zeta – functions for the random “pseudo-primes” (in the spirit of Cramer) have no analytic continuation through the critical line Re (z) = 1/2.

Viewed forwards in time, a population reproducing according to some random mechanism can be thought of as a branching process. What if it is viewed backwards? We can take a sample of individuals from the current generation and trace their genealogy backwards, and for instance find their most recent common ancestor; this is known as a coalescent process. If we know a population's random mating process, but have no actual data as to what the phylogenetic tree looks like, how do we derive the distribution of the time until its most recent common ancestor? I will discuss a variant on the classical Wright-Fisher reproductive model and deduce some parameter thresholds for emergence of different qualitative features of the tree. An isomorphic problem may also be useful in computer science for bounding the running time of certain random sampling algorithms.

We discuss methods for computing supports of degenerate diffusion processes. We assume throughout that the diffusion satisfies a stochastic differential equation on R^d whose drift vector field X₀ is ``polynomial'' and whose noise coefficients are constant. The case when each component of X₀ is of odd degree is well understood. Hence we focus our efforts on X₀ having at least one or more components of even degree. After developing methods to handle such cases, we shall apply them to specific examples, e.g. the Galerkin truncations of the Stochastic Navier-Stokes equation, to help establish ergodic properties of the resulting diffusion. One benefit to our approach is that, to prove such consequences, all we must do is compute certain Lie brackets.

Stochastic processing networks (SPNs) are a significant generalization of conventional queueing networks that allow for flexible scheduling through dynamic sequencing and alternate routing. SPNs arise naturally in a variety of applications in operations management and their control and analysis present challenging mathematical problems. One approach to these problems, via approximate diffusion control problems, has been outlined by J. M. Harrison. Various aspects of this approach have been developed mathematically, including a reduction in dimension of the diffusion control problem. However, other aspects have been less explored, especially, solution of the diffusion control problem, derivation of policies by interpretating such solutions, and limit theorems that establish optimality of such policies in a suitable asymptotic sense. In this talk, for a concrete class of networks called parallel server systems which arise in service network and computer science applications, we explore previously undeveloped aspects of Harrison's scheme and illustrate the use of the approach in obtaining simple control policies that are nearly optimal. Identification of a graphical structure for the network, an invariance principle and properties of local times of reflecting Brownian motion, will feature in our analysis. The talk will conclude with a summary of the current status and description of open problems associated with the further development of control of stochastic processing networks. This talk will draw on aspects of joint work with M. Bramson, M. Reiman, W. Kang and V. Pesic.

Consider a connected network and suppose each edge in the network has a random positive edge weight. Understanding the structure and weight of the shortest path between nodes in the network is one of the most fundamental problems studied in modern probability theory and goes under the name first passage percolation. It arises as a fundamental building block in many interacting particle system models such as the spread of epidemics on networks. To a large extent such problems have been only studied in the context of the n-dimensional lattice. In the modern context these problems take on an additional significance with the minimal weight measuring the cost of sending information while the number of edges on the optimal path (hopcount) representing the actual time for messages to get between vertices in the network. Given general models of random graphs with random edge costs, can one develop techniques to analyze asymptotics of functionals of interest which are robust to the model formulation? The aim of this talk is to describe a heuristic based on continuous time branching processes which gives very easily, a wide array of asymptotic results for random network models in terms of the Malthusian rate of growth and the stable age distribution of associated branching process. These techniques allow us to solve not only first passage percolation problems rigorously but also understand functionals such as the degree distribution of shortest path trees, congestion across edges as well as asymptotics for “betweeness centrality” a concept of crucial interest in social networks, in terms of Cox processes and extreme value distributions. These techniques also allow one to exactly solve models of “weak disorder” in the context of the stochastic mean field model of distance, a model of great interest in probabilistic combinatorial optimization.

We investigate an example of noise-induced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell,Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the addition of additive noise in the vertical direction leads to a unique invariant probability measure to which the system converges at a uniform, exponential rate. These facts are established primarily through the construction of a Lyapunov function which we generate as the solution to a sequence of Poisson equations. Unlike a number of other works, however, our Lyapunov function is constructed in a systematic way, and we present a meta-algorithm we hope will be applicable to other problems. We conclude by proving positivity properties of the transition density by using Malliavin calculus via some unusually explicit calculations. arXiv:1111.175v1 [math.PR]

The multi-population replicator dynamics (RD) can be considered a dynamic approach to the study of multi-player games, where it was shown to be related to Cross-learning, as well as of systems of co-evolving populations. However, not all of its equilibria are Nash equilibria (NE) of the underlying game, and neither convergence to an NE nor convergence in general are guaranteed. Although interior equilibria are guaranteed to be NE, no interior equilibrium can be asymptotically stable in the multi-population RD, resulting, e.g., in cyclic orbits around a single interior NE. We report on our investigation of a new notion of equilibria of RD, called mutation limits, which is based on the inclusion of a naturally arising, simple form of mutation, but is invariant under the specific choice of mutation parameters. We prove the existence of such mutation limits for a large range of games, and consider an interesting subclass, that of attracting mutation limits. Attracting mutation limits are approximated by asymptotically stable equilibria of the (mutation-)perturbed RD, and hence, offer an approximate dynamic solution of the underlying game, especially if the original dynamic has no asymptotically stable equilibria. Therefore, the presence of mutation will indeed stabilise the system in certain cases and make attracting mutation limits near-attainable. Furthermore, the relevance of attracting mutation limits as a game theoretic equilibrium concept is emphasised by the relation of (mutation-)perturbed RD to the Q-learning algorithm in the context of multi-agent reinforcement learning. However, in contrast to the guaranteed existence of mutation limits, attracting mutation limits do not exist in all games, raising the question of their characterization.

The Ising-Kac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighbourhood of radius $\ga^{-1}$ for $\ga \ll1$ around its base point. We study the Glauber dynamics for this model on a discrete two-dimensional torus $\Z^2/ (2N+1)\Z^2$, for a system size $N \gg \ga^{-1}$ and for an inverse temperature close to the critical value of the mean field model. We show that the suitably rescaled coarse-grained spin field converges in distribution to the solution of a non-linear stochastic partial differential equation. This equation is the dynamic version of the $\Phi^4_2$ quantum field theory, which is formally given by a reaction diffusion equation driven by an additive space-time white noise. It is well-known that in two spatial dimensions, such equations are distribution valued and a \textit{Wick renormalisation} has to be performed in order to define the non-linear term. Formally, this renormalisation corresponds to adding an infinite mass term to the equation. We show that this need for renormalisation for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value. This is a joint work with J.C. Mourrat (Lyon).

In this talk I will review the basic ideas of the regularity structure theory developed by Martin Hairer, as well as its applications to stochastic PDE problems. I will then discuss my joint work with Hairer on the sine-Gordon equation and central limit theorems for stochastic PDEs.

I will discuss a series of results concerning the effect of the derivative operator on the locations of the zeros of a random analytic function. Two models are considered. In the first, the zeros are chosen IID from some measure on the complex plane. In the second, the zeros are chosen to be a Poisson point process on the real line. Repeated differentiation results in a nearly deterministic zero set.

Consider a Galton-Watson tree. Pick two individuals at random by simple random sampling from the nth generation and trace heir lines of descent back in time till they meet. Call that generation X_n. In this talk we will discuss the probability distribution of X_n and its limits for the four cases m <1, m=1, m greater than 1 but finite, and m infinite, where m is the mean offspring size.

In this talk I will discuss a framework for transforming one variant of the Schramm–Loewner evolution (SLE) into another. The main tool in this approach is the Brownian loop measure. A simple case is to relate the reversal of radial SLE to whole-plane SLE, which looks the same locally. Writing the formula one might naïvely expect fails, because the loop measure term is infinite. In joint work with Greg Lawler, we show that there is a finite normalized version of the loop measure term, and that with this change, the naïve formula relating the two SLEs becomes correct.

Markov chain methods for Monte Carlo simulation of complex physical or statistical models often require significant tuning. Recent theoretical progress has renewed interest in "adaptive" Markov chain algorithms which learn from their sample history. However, these algorithms produce non-Markovian, time-inhomogeneous, irreversible stochastic processes, making rigorous analysis challenging. We show that lower bounds on the mixing times of these processes can be obtained using familiar ideas of hitting times and conductance from the theory of reversible Markov chains. The bounds obtained are sufficient to demonstrate slow mixing of several recently proposed algorithms including adaptive Metropolis kernels and the equi-energy sampler on some multimodal target distributions. These results provide the first non-trivial bounds on the mixing times of adaptive MCMC samplers, and suggest a way of classifying adaptive schemes that leads to new hybrid algorithms. Many open problems remain.

In this talk we study the rate of growth of the expectation of the number of holes (the rank of the first homology group) in a polyomino with uniform and percolation distributions. We prove the existence of linear bounds for the expected number of holes of a polyomino with respect to both the uniform and percolation distributions. Furthermore, we exhibit particular constants for the upper and lower bounds in the uniform distribution case. This results can be extend, using the same techniques, to other polyforms and higher dimensions.