We discuss three different proofs of the central limit theorem for the KPZ equation on a torus. All proofs rely on the geometric ergodicity of the so-called projective process, which, in this case, is the endpoint distribution of the corresponding directed polymer. I will try to draw connections to the subjects of homogenization, Malliavin calculus and the product of random matrices. I will also discuss what happens if we enlarge the torus as time increases.

A system of stochastic differential equations models the stock market, with coefficients depending on size of a stock relative to a benchmark. We fit the model using real-world financial data, and study long-term stability. This article is published in 2021 in Annals of Finance. We continue this research with current PhD students.

I will discuss a two-dimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. I will discuss how pointwise statistics of this equation can be related to a forward-backward stochastic differential equation (FBSDE) depending on the nonlinearity, as long as the nonlinearity is such that the FBSDE can be solved for long enough with Lipschitz decoupling function. I will also discuss conditions on the nonlinearity for this property to hold, and explain several cases in which the FBSDE can be solved explicitly. No previous knowledge of forward-backward SDEs will be assumed. This talk will be based on current joint work with Cole Graham and older joint work with Yu Gu.

This talk has two goals. The first is the derivation of a time-dependent KPZ equation (TDKPZ) from a time-inhomogeneous Ginzburg-Landau model, which describes, among many other things, stochastic dynamics of ferromagnets whose physics have been known to be time-dependent (due to Eddy currents) since the 1800s. The TDKPZ has a nonlinear twist that is not seen in the usual KPZ equation, making it a more interesting SPDE. The second goal is universality of the method for deriving (TD)KPZ, which should work beyond Ginzburg-Landau. In particular, we answer a question of deriving (TD)KPZ from asymmetric particle systems under natural fluctuation-versions of the assumptions in Yau's relative entropy method; the only additional assumption we need is a log-Sobolev inequality. Time permitting, future directions (of pure and applied flavors) will be discussed.

We will discuss stochastic quantization of the Yang-Mills model on two and three dimensional torus. In stochastic quantization we consider the Langevin dynamic for the Yang-Mills model which is described by a stochastic PDE. We construct local solution to this SPDE and prove that the solution has a gauge invariant property in law, which then defines a Markov process on the space of gauge orbits. We will also describe the construction of this orbit space, on which we have well-defined holonomies and Wilson loop observables. Based on joint work with Ajay Chandra, Ilya Chevyrev, and Martin Hairer.

The linear fractional stable motion generalizes two prominent classes of stochastic processes, namely stable Lévy processes, and fractional Brownian motion. For this reason it may be regarded as a basic building block for continuous time models. We study a stylized model consisting of a superposition of independent linear fractional stable motions, which nests various processes encountered in high-frequency financial economentrics as special cases. Applying an estimating equations approach, we construct estimators for the whole set of parameters of the mixed fractional stable motion, and we derive asymptotic normality in a high-frequency regime. The conditions for consistency turn out to be sharp for two prominent special cases: (i) for Lévy processes, i.e. for the estimation of the successive Blumenthal-Getoor indices, and (ii) for the mixed fractional Brownian motion introduced by Cheridito. In the remaining cases, our results reveal a delicate interplay between the Hurst parameters and the indices of stability. This talk is based on joint work with Mark Podolskij.

In recent years, there has been substantive empirical evidence that stochastic volatility is rough. In other words, the local behavior of stochastic volatility is much more irregular than semimartingales and resembles that of a fractional Brownian motion with Hurst parameter H < 0.5. In this paper, we derive a consistent and asymptotically mixed normal estimator of H based on high-frequency price observations. In contrast to previous works, we work in a semiparametric setting and do not assume any a priori relationship between volatility estimators and true volatility. Furthermore, our estimator attains a rate of convergence that is known to be optimal in a minimax sense in parametric rough volatility models.

This talk is based on joint work with Marc Hoffmann (Paris Dauphine), Yanghui Liu (Baruch College), Mathieu Rosenbaum and Grégoire Szymanski (both Ecole Polytechnique).

Hypergraphs capture multi-way relationships in data, and they have consequently seen a number of applications in higher-order network analysis, computer vision, geometry processing, and machine learning. We develop theoretical foundations for studying the space of hypergraphs using ingredients from optimal transport. By enriching a hypergraph with probability measures on its nodes and hyperedges, as well as relational information capturing local and global structures, we obtain a general and robust framework for studying the collection of all hypergraphs. First, we introduce a hypergraph distance based on the co-optimal transport framework of Redko et al. and study its theoretical properties. Second, we formalize common methods for transforming a hypergraph into a graph as maps between the space of hypergraphs and the space of graphs, and study their functorial properties and Lipschitz bounds. Finally, we demonstrate the versatility of our Hypergraph CoOptimal Transport (HyperCOT) framework through various examples. If time permits, I will discuss the application of optimal transport in scientific data visualization

Canceled

Several well-known problems in first-passage percolation relate to the behavior of infinite geodesics: whether they coalesce and how rapidly, and whether doubly infinite ``bigeodesics'' exist. In the plane, a version of coalescence of ``parallel'' geodesics has previously been shown; we will discuss new results that show infinite geodesics from the origin have zero density in the plane. We will describe related forthcoming work showing that geodesics coalesce in dimensions three and higher, under unproven assumptions believed to hold below the model's upper critical dimension. If time permits, we will also discuss results on the bigeodesic question in dimension three and higher.

In first-passage percolation (FPP), we let $\tau_v$ be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If $F$ is the distribution function of $\tau_v$, there are different regimes: if $F(0)$ is small, this weight typically grows like a linear function of the distance, and when $F(0)$ is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results that show that if the sum of $F^{-1}(1/2+1/2^k)$ diverges, then a.s. there are exceptional times at which the weight grows atypically, but if sum of $k^{7/8} F^{-1}(1/2+1/2^k)$ converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. Then we will consider what the model looks like when passage times are unusually small by considering an analogous construction to Kesten's incipient infinite cluster in the FPP setting. This is joint work with M. Damron, J. Hanson, W.-K. Lam.

TBA