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.

We study the long time behaviour of the heat kernel on Abelian covers of compact Riemannian manifolds. For manifolds without boundary work of Lott and Kotani-Sunada establishes precise long time asymptotics. Extending these results to manifolds with boundary reduces to a cute eigenvalue minimization problem, which we resolve for a Dirichlet and Neumann boundary conditions. We will show how these results can be applied to studying the ``winding'' / ``entanglement'' of Brownian trajectories in Riemannian manifolds.

The idea behind excited random walks (ERWs), roughly speaking, is to take a well-known underlying process (such as, for example, simple symmetric random walk (SSRW)) and modify its transition probabilities for the "first few" visits to every site of the state space. These modifications can be deterministic or random. The resulting process is not markovian, and its properties can be very different from those of the underlying process. I shall give a short review of some of the known results for ERW (with SSRW as underlying process) on the d-dimensional integer lattice and then concentrate on a specific model for d=1. For this model we can give a complete picture including functional limit theorems.

We consider a Bayesian framework for making inferences about dynamical systems from ergodic observations. The proposed Bayesian procedure is based on the Gibbs posterior, a decision theoretic generalization of standard Bayesian inference. We place a prior over a model class consisting of a parametrized family of Gibbs measures on a mixing shift of finite type. This model class generalizes (hidden) Markov chain models by allowing for long range dependencies, including Markov chains of arbitrarily large orders. We characterize the asymptotic behavior of the Gibbs posterior distribution on the parameter space as the number of observations tends to infinity. In particular, we define a limiting variational problem over the space of joinings of the model system with the observed system, and we show that the Gibbs posterior distributions concentrate around the solution set of this variational problem. In the case of properly specified models our convergence results may be used to establish posterior consistency. This work establishes tight connections between Gibbs posterior inference and the thermodynamic formalism, which may inspire new proof techniques in the study of Bayesian posterior consistency for dependent processes.

Consider a crystal formed of two types of atoms placed at the nodes of the integer lattice. The type of each atom is chosen at random, but the crystal is statistically shift-invariant. Consider next an electron hopping from atom to atom. This electron performs a random walk on the integer lattice with randomly chosen transition probabilities (since the configuration seen by the electron is different at each lattice site). This process is highly non-Markovian, due to the interaction between the walk and the environment. We will present a martingale approach to proving the invariance principle (i.e. Gaussian fluctuations from the mean) for (irreversible) Markov chains and show how this can be transferred to a result for the above process (called random walk in random environment). This is joint work with Timo Sepp\"al\"ainen.

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.

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 introduce a new measure constructed from high-frequency financial data which we call the Realized Laplace Transform of volatility. The statistic provides a nonparametric estimate for the empirical Laplace transform of the latent stochastic volatility process over a given interval of time. When a long span of data is used, i.e., under joint long-span and fill-in asymptotics, it is an estimate of the volatility Laplace transform. The asymptotic behavior of the statistic depends on the small scale behavior of the driving martingale. We derive the asymptotics both in the case when the latter is known and when it needs to be inferred from the data. When the underlying process is a jump-diffusion our statistic is robust to jumps and when the process is pure-jump it is robust to presence of less active jumps. We apply our results to simulated and real financial data.

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.

The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.

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.

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.

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.

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.

The classical result of Bramson gives a precise logarithmic correction to the speed of front propagation in one dimensional branching random walks and Brownian motions. I will discuss several variants of this model where the slowdown term is not classical.

I'll discuss the Arzela-Ascoli theorem in the Skorokhod space D(E) of E-valued functions that are right-continuous with left limits. There are several topologies one uses on D(E). For the most common J_1-topology, a version of the Arzela-Ascoli theorem is standard for quite general spaces E. In the weaker and less used M_1-topology however, a version of the theorem has only been available when E is a vector space. I'll describe a generalization to the setting where E is the metric space of finite Borel measures on the real line. I'll also show an application from a recent queueing theory result, where M_1 is the most natural choice of topology.

Motivated by several biological questions, including insect respiration, we consider ODEs with stochastically switching right-hand sides and PDEs with stochastically switching boundary conditions. In a variety of situations, we prove that the system exhibits surprising behavior. In this talk, we will highlight some of the most interesting results and describe their implications both for the mathematical study of stochastic hybrid systems and for insect respiration.

Branching random walk can be viewed as particles performing random walks while branching at integer time. We review some of the existing results on the maximal (or minimal) displacement, when each particle moves and branches independently according the same step distribution and the same branching law. Then we will compare them with similar but different models. Roughly speaking, in one variation, we will consider the asymptotic behavior of the particle at time n, whose ancestors’ location are consistently small. In another variation, we will consider the maximal displacement for the model, where the step distributions vary with respect to time.

We study a random walk on the integers Z which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, p and q. R consecutive right jumps from a site in the q-mode are required to switch it to the p-mode, and L consecutive left jumps from a site in the p-mode are required to switch it to the q-mode. From a site in the p-mode the walk jumps right with probability p and left with probability (1-p), while from a site in the q-mode these probabilities are q and (1-q). We prove a sharp cutoff for right/left transience of the random walk in terms of an explicit function of the parameters $\alpha = \alpha(p,q,R,L)$. For $\alpha > 1/2$ the walk is transient to $+\infty$ for any initial environment, whereas for $\alpha < 1/2$ the walk is transient to $-\infty$ for any initial environment. In the critical case, $\alpha = 1/2$, the situation is more complicated and the behavior of the walk depends on the initial environment. We are able to give a characterization of transience/recurrence in many instances, including when either R=1 or L=1 and when R=L=2. In the noncritical case, we also show that the walk has positive speed, and in some situations are able to give an explicit formula for this speed. This is joint work with my former post-doc, Nick Travers, now at Indiana University.

We introduce some recent advances in the study of nonlinear stochastic heat equations, and related stochastic PDEs. Special attention will be paid to the local structure of the solution. In particular, we show that, frequently, the solution exhibits a form of intermittency. Time permitting, we discuss related connections to classical potential theory and mathematical physics as well.

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.

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.

We introduce the term `superconcentration' to describe the phenomenon when a function of a Gaussian random field exhibits a far stronger concentration than predicted by classical concentration of measure. We show that when superconcentration happens, the field becomes chaotic under small perturbations and a `multiple valley picture' emerges. Conversely, chaos implies superconcentration. While a few notable examples of superconcentrated functions already exist, e.g. the largest eigenvalue of a GUE matrix, we show that the phenomenon is widespread in physical models; for example, superconcentration is present in the Sherrington-Kirkpatrick model of spin glasses, directed polymers in random environment, the Gaussian free field and the Kauffman-Levin model of evolutionary biology. As a consequence we resolve the long-standing physics conjectures of disorder-chaos and multiple valleys in the Sherrington-Kirkpatrick model, which is one of the focal points of this talk.

Evolution by natural selection can act at multiple biological levels, often in opposing directions. This is particularly the case for pathogen evolution, which occurs both within the host it infects and via transmission between hosts, and for the evolution of cooperative behavior, where individually advantageous strategies are disadvantageous at the group level. In mathematical terms, these are multiscale systems characterized by stochasticity at each scale. We show how a simple and natural formulation of this can be viewed as a ball-and-urn (measure-valued) process. This equivalent process has very nice mathematical properties, namely it converges weakly to either (i) the solution of an analytically tractable integro-partial differential equation or (ii) a Fleming-Viot process. We can then study properties of these limiting objects to infer general properties of multilevel selection.

In this talk, I will discuss several finite-horizon Markov decision problems (MDPs) in which the goal is to gather distributional information regarding the total reward that one obtains when implementing a policy that maximizes total expected rewards. I will begin by studying the optimal policy for the sequential selection of an alternating subsequence from a sequence of n independent observations from a continuous distribution, and I will prove a central limit theorem for the number of selections made by that policy. Then, I will discuss a simple version of a sequential knapsack problem, and I will use its structure to characterize a class of MDPs in which the optimal total reward has variance that can be bounded in terms of its mean. Surprisingly, such characterization turns out to be common in several examples of MDPs from operations research, financial engineering and combinatorial optimization. (The talk is based on joint work with Robert W. Chen, Noah Gans, Larry Shepp, and J. Michael Steele.)

Talk given at SEPC 2021 in honor of Elizabeth Meckes