How does intrinsic noise from stochastic ion channels affect spontaneous activity in a single neuron? The size of a neuron affects how many ion channels in the membrane facilitate the generation of action potentials. Fewer ion channels causes an increase in the number of spontaneous action potentials. The density of neural tissue is therefore fundamentally limited by spontaneous activity. Evolutionary pressure tends to favor neural tissue with higher density for a variety of reasons, but the increase in spontaneous activity limits how dense it can become and still remain functional. I will talk about how large deviation theory can be used to quantify the relationship between ion channels (type and number) and spontaneous activity.

A popular family of models of random partitions is called the Chinese Restaurant Process. We imagine n customers being seated randomly and sequentially at tables of a restaurant according to a fixed stochastic rule. Grouping customers by the tables gives us a partition of n. Consider a Markov chain on such partitions where we remove a randomly chosen customer and reseat her. How can one describe the limit of such a Markov chain as n tends to infinity? We will construct such limits as diffusions on partitions of the unit interval. Examples of such random partitions of the unit interval are given by the complement of the zeros of the Brownian motion or the Brownian bridge. The processes of ranked interval lengths of our partitions are members of a family of diffusions introduced by Ethier and Kurtz (1981) and Petrov (2009) that are stationary with respect to the Poisson-Dirichlet distributions. Our construction is a piece of a more complex diffusion on the space of real trees, stationary with respect to the law of the Brownian Continuum Random Tree, whose existence has been conjectured by David Aldous. Joint work with Noah Forman, Doug Rizzolo, and Matthias Winkel.

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.

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.

I will present a few recent results on random planar metrics of two-dimensional discrete Gaussian free fields, including Liouville first passage percolation, the chemical distance for level-set percolation and the electric effective resistance on an associated random network. Besides depicting a fascinating picture for 2D GFF, these metric aspects are closely related to various models of planar random walks.

For several classical matrix models the joint density of the eigenvalues can be written as an expression involving a Vandermonde determinant raised to the power of 1, 2 or 4. Most of these examples have beta-generalizations where this exponent is replaced by a parameter beta>0. In recent years the point process limits of various beta ensembles have been derived. The limiting processes are usually described as the spectrum of certain stochastic operators or with the help of a coupled system of SDEs. In the bulk beta Hermite case (which is the generalization of GUE) there is a nice geometric construction of the point process involving a Brownian motion in the hyperbolic plane, this is the Brownian carousel. Surprisingly, there are a number of other limit processes that have carousel like representation. We will discuss a couple of examples and some applications of these new representations. Joint with Balint Virag.

In this talk we consider second-order finite Markov chains that are > trajectorially reversible, a property that is a generalization of the > notion of reversibility for usual Markov chains. Specifically, we > study spectral properties of second-order Markov chains that have a > tendency to not return to their previous state. We confirm that > resorting to second-order chains can be an option to improve the speed > of convergence to equilibrium. This is joint work with Persi Diaconis > and Laurent Miclo.

A Dirichlet random probability \(P_t\) on \(\mathbb{R}^d\) of intensity \(t\) and governed by the probability \(\alpha\) is such that for any partition \( (A_0,\ldots,A_n)\) of \(\mathbb{R}^d\) the random variable \( (P_t(A_0),\ldots,P_t(A_n))\) is Dirichlet distributed with parameters \( (t\alpha(A_0),\ldots,t\alpha(A_n).\) If \(\mu(t\alpha)\) is the distribution of \(X_t=\int xP_t(dx),\) the Dirichlet curve is the map \(t\mapsto \mu(t\alpha)\). Its study raises challenging problems and explicit computations are rare. We prove that if \(\lim_{t\to\infty}\mu(t\alpha)\) exists, it is a Cauchy or Dirac distribution on \(\mathbb{R}^d\). If \(\alpha\) has an expectation we prove that \(t\mapsto \int \psi(x)\mu(t\alpha)(dx)\) is decreasing for any positive convex function \(\psi\) on \(\mathbb{R}^d.\) In other terms the Dirichlet curve decreases in the Strassen order. This is joint work with Mauro Piccioni.

Many applications can be modeled as a large system of homogeneous interacting particles on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle.

We will discuss recent progress, jointly with R. Gheissari, on the dynamical phase transition for the critical q-state Potts model on the 2D torus (both single-site dynamics such as Glauber/Metropolis and cluster dynamics such as Swendsen--Wang), where the conjectured behavior was a mixing time that is polynomial in the side-length for $q = 2,3,4$ colors but exponential in it for $q>4$. We will then present a proof from a recent work with R. Gheissari and Y. Peres, that, on the complete graph on $n$ vertices with $q>2$ colors, the Swendsen--Wang dynamics is exponentially slow in $n$, improving on the lower bound of $\exp(c\sqrt{n})$ due to Gore and Jerrum in 1999. If time permits, we will then revisit the model on the 2D lattice, and describe the effect of different boundary conditions on its dynamical behavior at criticality.

I will discuss a class of particle systems that serve as models for supercooling in physics, neuron firing in neuroscience and systemic risk in finance. The interaction between the particles falls into the mean-field framework pioneered by McKean and Vlasov in the late 1960s, but many new phenomena arise due to the singularity of the interaction. The most striking of them is the loss of regularity of the particle density caused by the the self-excitation of the system. In particular, while initially the evolution of the system can be captured by a suitable Stefan problem, the following irregular behavior necessitates a more robust probabilistic approach. Based on joint work with Sergey Nadtochiy.

Recent experiments show that an overdamped Brownian particle in a diffusion gradient experiences an additional drift. Equivalently, the Langevin equation describing the particle's motion should be interpreted according to the "anti-Ito" definition of stochastic integrals. I will explain this effect mathematically by studying the zero-mass limit of the stochastic Newton's equation modeling the particle's motion and, using a multiscale expansion, extend the analysis to a wide class of equations, including systems with colored noise and delay terms, interpreting recent electrical circuit experiments. The results were obtained in a collaboration with experimental physicists in Stuttgart: Giovanni Volpe, Clemens Bechinger, Laurent Helden and Thomas Brettschneider, as well as with the mathematics graduate students at the University of Arizona: Scott Hottovy and Austin McDaniel.

The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this work we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness, and tightness) of certain perturbations of the original process. This is a joint work with P.Dupuis and V.Maroulas.

In this talk we present a recent result on the smoothness of the density for the solution of a semilinear heat equation with multiplicative space-time Gaussian white noise. We assume that the coefficients are smooth and the diffusion coefficient is not identically zero at the initial time. The proof of this result is based on the techniques of the Malliavin calculus, and the existence of negative moments for the solution of a linear heat equation with multiplicative space-time white noise.

We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian in the regime of small noise. We give a new proof of the Eyring-Kramers formula for the spectral gap of the associated generator of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg and of the mean-difference estimate introduced by Chafai and Malrieu. The Eyring-Kramers formula follows as a simple corollary from two main ingredients : The first one shows that the Gibbs measures restricted to a domain of attraction has a "good" Poincaré constant mimicking the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of the mean-difference by a new weighted transportation distance. It contains the main contribution of the spectral gap, resulting from exponential long waiting times of jumps between metastable states of the diffusion. This new approach also allows to derive sharp estimates on the log-Sobolev constant.

Parking functions were introduced by combinatorialists in the 1960s, and have recently been studied by probabilists. When the parking lot is an infinite graph and cars drive around at random, we will look at how many parking spots are needed for every car to eventually find a spot.

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.

Consider a network of sites growing over time such that at step n a newcomer chooses a vertex from the existing vertices with probability proportional to a function of the degree of that vertex, i.e., the number of other vertices that this vertex is connected to. This is called a preferential attachment random graph. The objects of interest are the growth rates for the growth of the degree for each vertex with n and the behavior of the empirical distribution of the degrees. In this talk we will consider three cases: the weight function w(.) is superlinear, linear, and sublinear. Using recently obtained limit theorems for the growth rates of a pure birth continuous time Markov chains and an embedding of the discrete time graph sequence in a sequence of continuous time pure birth Markov chains, we establish a number of results for all the three cases. We show that the much discussed power law growth of the degrees and the power law decay of the limiting degree distribution hold only in the linear case, i.e., when w(.) is linear

The frog model is a type of branching random walk model. Active "frogs" move according to random walks, and if they encounter a sleeping frog on their walk, the sleeping frog becomes active and begins an independent random walk. Over the past 20 years, recurrence properties and asymptotic behavior of this system (and many generalizations) have been studied extensively. One way to generalize this system is to consider the continuous version: Brownian motion frogs moving in R^d. In this talk, we will describe a continuous variant of the problem and show a limiting shape theorem analogous to prior discrete results.

Coupling is a way of constructing Markov processes with prescribed laws on the same space. The coupling is called Markovian if the coupled processes are co-adapted to the same filtration. We will first investigate Markovian couplings of elliptic diffusions and demonstrate how the rate of coupling (how fast you can make the coupled processes meet) is intimately connected to the geometry of the underlying space. Next, we will consider couplings of hypoelliptic diffusions (diffusions driven by vector fields whose Lie algebra span the whole tangent space). Constructing successful couplings (where the coupled processes meet almost surely) for these diffusions is a much more subtle question as these require simultaneous successful coupling of the driving Brownian motions as well as a collection of their path functionals. We will construct successful Markovian couplings for a large class of hypoelliptic diffusions. We will also investigate non-Markovian couplings for some hypoelliptic diffusions, namely the Kolmogorov diffusion and Brownian motion on the Heisenberg group, and demonstrate how these couplings yield sharp estimates for the total variation distance between the laws of the coupled diffusions when Markovian couplings fail. Furthermore, we will demonstrate how non-Markovian couplings can be used to furnish purely analytic gradient estimates of harmonic functions on the Heisenberg group by purely probabilistic means, providing yet another strong link between probability and geometric analysis. This talk is based on joint works with Wilfrid Kendall, Maria Gordina and Phanuel Mariano.

A link to the paper can be found on her web page. In solid tumors, targeted treatments can lead to dramatic regressions, but responses are often short-lived because resistant cancer cells arise. The major strategy proposed for overcoming resistance is combination therapy. We present a mathematical model describing the evolutionary dynamics of lesions in response to treatment. We first studied 20 melanoma patients receiving vemurafenib. We then applied our model to an independent set of pancreatic, colorectal, and melanoma cancer patients with metastatic disease. We find that dual therapy results in long-term disease control for most patients, if there are no single mutations that cause cross-resistance to both drugs; in patients with large disease burden, triple therapy is needed. We also find that simultaneous therapy with two drugs is much more effective than sequential therapy. Our results provide realistic expectations for the efficacy of new drug combinations and inform the design of trials for new cancer therapeutics.

We study N-player repeated Cournot competitions that model the determination of price in an oligopoly where firms choose quantities. These are nonzero-sum (ordinary and stochastic) differential games, whose value functions may be characterized by systems of nonlinear Hamilton-Jacobi-Bellman partial differential equations. When the quantity being produced is in finite supply, such as oil, exhaustibility enters as boundary conditions for the PDEs. We analyze the problem when there is an alternative, but expensive, resource (for example solar technology for energy production), and give an asymptotic approximation in the limit of small exhaustibility. We illustrate the two-player problem by numerical solutions, and discuss the impact of limited oil reserves on production and oil prices in the dupoly case. Joint work with Chris Harris (Cambridge University) and Sam Howison (Oxford University).

An open problem in Classical Differential Geometry, posed by S. T. Yau, asks when does a simple closed curve in Euclidean 3-space bound a surface of positive curvature? We will give a survey of recent results related to this problem, including connections with the h-principle, Monge-Ampere equations, and Alexandrov spaces with curvature bounded below. In particular we will discuss joint work with Stephanie Alexander and Jeremy Wong on Topological finiteness theorems for nonnegatively curved surfaces filling a prescribed boundary, which use in part the finiteness and stability theorems of Gromov and Perelman.

Gaussian processes are employed to model spatially varying errors in various stochastic systems. In this talk, we consider the analysis of the extreme behaviors and the rare-event simulation problems for such systems. In particular, the topic covers various nonlinear functionals of Gaussian processes including the supremum norm and integral of convex functions. We present the asymptotic results and the efficient simulation algorithms for the associated rare-event probabilities.

The theory of large deviations has been used in the development of Monte Carlo methods for estimating quantities defined in terms of a specific rare event, such as ruin probabilities or buffer overflow probabilities. However, rare events also play an important role when estimating functionals of an invariant distribution, where straightforward simulation will converge very slowly when parts of the state space do not communicate well. Problems of this sort are common in statistical inference, engineering and the physical sciences. After reviewing some of the methods used to accelerate the convergence of Monte Carlo, we consider the use of the large deviation rate for the empirical measure as a performance measure and introduce a new class of algorithms (which we call infinite swapping schemes) that optimize this rate.

The superprocess or Dawson-Watanabe process is one of the most intensively studied stochastic processes of the last quarter century. It arises as a limit of population processes, and includes information about the physical location of individuals. Usually the superprocess is measure valued, but In one dimension it has a density that satisfies a parabolic stochastic PDE. For a long time uniqueness for this equation was unknown. In joint work with Barlow, Mytnik, and Perkins, we show that nonuniquess holds for the superprocess equation and several related equations.