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.

Pratima Hebbar, Probability Seminar on October 21, 2021

Data lying in a high dimensional ambient space are commonly thought to have a much lower intrinsic dimension. In particular, the data may be concentrated near a lower-dimensional subspace or manifold. There is an immense literature focused on approximating the unknown subspace and the unknown density, and exploiting such approximations in clustering, data compression, and building of predictive models. Most of the literature relies on approximating subspaces and densities using a locally linear, and potentially multiscale, dictionary with Gaussian kernels. In this talk, we propose a simple and general alternative, which instead uses pieces of spheres, or spherelets, to locally approximate the unknown subspace. I will also introduce a curved kernel called the Fisher–Gaussian (FG) kernel which outperforms multivariate Gaussians in many cases. Theory is developed showing that spherelets can produce lower covering numbers and mean square errors for many manifolds, as well as the posterior consistency of the Dirichlet process mixture of the FG kernels. Time permitting, I will also talk about an ongoing project about stochastic differential geometry.

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.

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.

Random Polymers are modeled as a one dimensional random walk (S_n), with excursion length distribution P(S_1 = n) = \phi(n)/n^\alpha, \alpha > 1 and \phi(n) a slowly varying function. The polymer gets a random reward whenever it visits or crosses an interface. The random rewards are realised as a sequence of i.i.d. variables (\omega_n). Depending on the relation between the mean value of the disorder \omega_n and the temperature, the polymer might prefer to stick to the interface (pinnings) or undergo a long excursion away from it (depinning). In this talk we will review some aspects of random polymer models. We will also discuss in more detail the pinning-depinning transition of the `Pinning' model and prove its annealed and quenched critical points are distinct. This is joint work with Ken Alexander.

This is a joint work with B.Driver. The groups in question are modeled on an abstract Wiener space. Then a group Brownian motion is defined, and its properties are studied in connection with the geometry of this group. The main results include quasi-invariance of the heat kernel measure, log Sobolev inequality (following a bound on the Ricci curvature), and the Taylor isomorphism to the corresponding Fock space. The latter is a version of the Ito-Wiener expansion in the non-commutative setting.

In this talk, I will discuss a generalized Moran process from the evolutionary game theory. The generalization incorporates arrangement of by graphs and games among individuals. For these additional features, there has been consistent interest in using general spatial structure as a way to explain the ubiquitous game behavior in biological evolutions; the introduction of games leads to technical complications as basic as nonlinearity and asymmetry in the model. The talk will be centered around a seminal finding in the evolutionary game theory that was obtained more than a decade ago. By an advanced mean-field method, it reduces the infinite-dimensional problem of solving for the game fixation probabilities to a one-dimensional diffusion problem in the limit of a large population. The recent mathematical results and some related mathematical methods will be explained.

Abstract:One way to convince ourselves that no cooperation can evolve among defectors is via a simple yet one of the most famous games in all of game theory - the Prisoner’s dilemma (PD) game. The players of this game adopt one of the two strategies: a) a cooperator who pays a cost so that another individual can receive a benefit, or b) a defector who can receive benefits, but it has no cost as it does not deal out any benefits at all. As seen from this formulation, no rational individual would opt to be a cooperator. Yet, we can see cooperation everywhere around us and thus (assuming defectors were here first) there must exist at least one mechanism for its evolution. Nowak (2006, 2012) discusses several of such mechanisms, including the kin selection by which cooperation can spread if the benefits go primarily to genetic relatives. In this talk we will introduce a simple PD-like asymmetric matrix game and show how Hamilton’s rule can easily be recovered. We will also introduce a simple PD-like symmetric matrix game to model the evolution of cooperation via greenbeard mechanism, which can be seen as a special case of kin selection.

The dynamics of a set of chemical reactions are usually modeled by mass action kinetics as a set of algebraic ordinary differential equations. This model sees the state space of the system as a continuum, whereas chemical reactions represent interactions of a discrete set of molecules. We study large fluctuations of the stochastic mass action kinetics model through Freidlin-Wentzell theory. The application of such a theory to this framework requires justification, in particular because of the non-uniformily Lipschitz character of the model. We therefore find, using tools of Lyapunov stability theory, a set of sufficient conditions for the applicability of large deviations theory to this framework, and prove that such conditions are satisfied by a large class of chemical reaction networks identified exclusively on the base of their topological structure.

Quantum field theory is the theoretical framework for studying fundamental interactions. "Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. The recent years witnessed interesting progress in understanding solutions of these stochastic PDEs, one of the remarkable examples being Hairer and Mourrat-Weber's results on the Phi^4_3 equation.

In this talk we will discuss stochastic quantization of quantum field theory with gauge symmetries, with focus on an Abelian example but also provide prospects of non-Abelian Yang-Mills theories. We address issues regarding Wilson’s lattice regularization, dynamical gauge fixing, renormalization, Ward identities, and construction of dynamical loop and string observables.

Integrable stochastic particle systems in one space dimension, like the Totally Asymmetric Simple Exclusion Process (TASEP), have been studied for over 50 years (introduced simultaneously in biology and mathematics in 1969-70). They strike a balance between being simple enough to be mathematically tractable and complicated enough to describe many interesting phenomena. Many natural questions about these systems can be generalized by introducing multiple parameters. The interplay between these parameters is powered by the Yang-Baxter equation, which brings new intriguing results to the well-traveled territory. In particular, I will discuss new Lax-type equations for the Markov semigroups of the TASEP and its relatives. Based on a joint work with Axel Saenz.

Directed polymers are a statistical mechanics model for random growth. Their partition functions are solutions to a discrete stochastic heat equation. This talk will discuss the logarithmic derivatives of the partition functions, which are solutions to a discrete stochastic Burgers equation. Of interest is the success or failure of the “one force-one solution principle” for this equation. I will reframe this question in the language of polymers, and share some surprising results that follow. Based on joint work with Louis Fan and Timo Seppäläinen.

Recently, Markov-chain Monte Carlo methods based on non-reversible piecewise deterministic Markov processes (PDMP) are under growing attention, thanks to the increase in performance they usually bring. Beyond their numerical efficacy, the non-reversible and piecewise deterministic characteristics of these processes prompt interesting questions, regarding for instance ergodicity proof and convergence bounds. During this talk, I will particularly focus on the obtained results and open problems left while considering PDMP evolution of particle systems, both in an equilibrium and out-of-equilibrium setting. Hardcore particle systems have embodied a testbed of choice since the first implementations of Markov chain Monte Carlo in the 50’s. Even today, the entropic barriers they exhibit are still resisting to the state-of-the-art MCMC sampling methods. During this talk, I will review the recent developments regarding sampling such systems and discuss the dynamical bottlenecks that are yet to be solved.

The six-vertex model, also known as the square-ice model, is one of the central and most studied systems of 2d statistical mechanics. It offers various combinatorial interpretations. One of them involves molecules of water on the square grid; another one deals with non-intersecting lattice paths, which can be also viewed as level lines of an integer-valued height function. Despite many efforts since the 1960s, the limit shapes for the height function are still unknown in general situations. However, we recently found ways to compute them in a degeneration, which leads to a low density of corners of paths (or, equivalently, of horizontal/vertical molecules of water). I will report on the progress in this direction emphasizing various unusual features: appearance of hyperbolic PDEs; discontinuities in densities; connections to random permutations.