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.

Talk given at SEPC 2021 in honor of Elizabeth Meckes

Talk given at SEPC 2021 in honor of Elizabeth Meckes

Branching Brownian motion is a classical process in probability theory belonging to the class of “Log-correlated random fields”. We study the structure of extreme level sets of this process, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida.(joint work with A. Cortines, O Louidor)

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.

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.

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.

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.

The goal of this talk is to further a joint project involving Carla Staver, Simon Levin, Rick Durrett, and Ruibo Ma. The puzzle is: why can savannah and forest display stable coexistence when this is not possible in a spatially homogeneous system.

In Markov chain Monte Carlo, a Markov chain is constructed whose limiting distribution is equal to some target distribution. While it is easy to build such chains, for some distributions the standard constructions can take exponentially long to come near that limit, making the chain torpidly mixing. When the limit is reached in polynomial time, the chain is rapidly mixing. Tempering is a technique designed to speed up the convergence of Markov chains by adding an extra temperature parameter that acts to smooth out the target distribution. In this talk I will present joint work with Dawn Woodard (Cornell) and Scott Schmidler (Duke) that give sufficient conditions for a tempering chain to be torpidly mixing, and a related (but different) set of conditions for the chain to be rapidly mixing.

We propose the following model of a random graph on $n$ vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair ij with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by picking a random point X from F and defining a graph on n vertices whose edges are pairs ij for which X_{ij} \le p. The standard Erd\H{o}s-R\'{e}nyi model is the special case when F is uniform on the 0-1 unit cube. We determine basic properties such as the connectivity threshold for quite general distributions. We also consider cases where the X_{ij} are the edge weights in some random instance of a combinatorial optimization problem. By choosing suitable distributions, we can capture random graphs with interesting properties such as triangle-free random graphs and weighted random graphs with bounded total weight. This is joint work with Alan Frieze (CMU) and Juan Vera (Waterloo). The talk will be self-contained and no prior knowledge of random graphs is assumed.

In this talk we present two intimately related approaches in speeding-up molecular simulations via Monte Carlo simulations. First, we discuss coarse-graining algorithms for systems with complex, and often competing particle interactions, both in the equilibrium and non-equilibrium settings, which rely on multilevel sampling and communication. Second, we address mathematical, numerical and algorithmic issues arising in the parallelization of spatially distributed Kinetic Monte Carlo simulations, by developing a new hierarchical operator splitting of the underlying high-dimensional generator, as means of decomposing efficiently and systematically the computational load and communication between multiple processors. The common theme in both methods is the desire to identify and decompose the particle system in components that communicate minimally and thus local information can be either described by suitable coarse-variables (coarse-graining), or computed locally on a individual processors within a parallel architecture.

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.

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.