Mathematics

​Stochastic (partial) differential equations with singular coefficients

​Longjie Xie (Jiangsu Normal University & UIUC)

Abstract: It is a classical result that the ordinary differential equation is well-posed for Lipschitz coefficient but usually ill-posed if the coefficient is only Holder continuous. However, this dramatically changes if the system is perturbed by a noise. The purpose of this talk is to give a brief introduction and overview of the topic of regularization and well-posedness by noise for ordinary and partial differential equations.

Effective modelling for some SPDEs

Wei Wang (Nanjing University & IIT)

Abstract: This talk introduce some effective modelling for some SPDEs with separated time scales. First we consider SPDEs with slow-fast part and averaging method is applied to derive the averaged approximation model. Normal deviation is also considered and large deviations further confirms the effectivity of the averaged equation plus the normal deviation. Second, we consider diffusion approximation for a Burgers type equation wit stochastic advection. By constructing a martingale, the approximation on both finite tie interval and infinite time interval is derived. Last the diffusion approximation is also applied to study the Smoluchowski-Kramers approximation for a nonlinear wave equation with state-dependent damping and random fluctuation.

Local density estimate for a hypoelliptic SDE

Cheng Ouyang (UIC Math)

Abstract: In a series of three papers in the 80’s, Kusuoka and Stroock developed a probabilistic program in order to obtain sharp bounds for the density function of a hypoelliptic SDE driven by a Brownian motion. We aim to investigate how their method can be used to study rough SDEs driven by fractional Brownian motions. In this talk, I will outline Kusuoka and Stroock’s approach and point out where the difficulties are in our current setting. The talk is based on an ongoing project with Xi Geng and Samy Tindel.

The size of the boundary in first-passage percolation

Wai-Kit Lam (Indiana University Bloomington)

Abstract: First-passage percolation is a random growth model defined using i.i.d. edge-weights $(t_e)$ on the nearest-neighbor edges of $\mathbb{Z}^d$. An initial infection occupies the origin and spreads along the edges, taking time $t_e$ to cross the edge $e$. In this talk, we study the size of the boundary of the infected region at time $t$, $B(t)$. Under a weak moment condition on the weights, we show that for most times, $\partial B(t)$ has size of order $t^{d-1}$ (smooth). On the other hand, for heavy-tailed distributions, $B(t)$ contains many small holes, and consequently we show that $\partial B(t)$ has size of order $t^{d-1+\alpha}$ for some $\alpha > 0$ depending on the distribution. In all cases, we show that the exterior boundary of $B(t)$ (edges touching the unbounded component of the complement of $B(t)$) is smooth for most times. Under the unproven assumption of uniformly positive curvature on the limit shape for $B(t)$, we show the inequality $\partial B(t) \leq (log t)^C t^{d-1}$ for all large t. This is a joint work with Michael Damron and Jack Hanson.

Random walks, Laplacians, and volumes in sub-Riemannian geometry

Robert Neel (Lehigh University)

Abstract: We study a variety of random walks on sub-Riemannian manifolds and their diffusion limits, which give, via their infinitesimal generators, second-order operators on the manifolds. A primary motivation is the lack of a canonical Laplacian in sub-Riemannian geometry, and thus we are particularly interested in the relationship between the limiting operators, the geodesic structure, and operators which can be obtained as divergences with respect to various choices of volume. This work is joint with Ugo Boscain (CNRS), Luca Rizzi (CNRS), and Andrei Agrachev (SISSA).

Estimates of Dirichlet heat kernels for subordinate Brownian motions

Panki Kim (Seoul National University)

Abstract: In this talk, we discuss estimates of transition densities of subordinate Brownian motions in open subsets of Euclidean space. When open subsets are $C^{1,1}$ domain, we establish sharp two-sided estimates for the transition densities of a large class of killed subordinate Brownian motions whose scaling order is not necessarily strictly below 2. Our estimates are explicit and written in terms of the dimension, the Euclidean distance between two points, the distance to the boundary and the Laplace exponent of the corresponding subordinator only. We also establish boundary Harnack principle in $C^{1,1}$ open set with explicit decay rate. This is a joint with with Ante Mimica.

On the potential theory of subordinate killed processes

Zoran Vondraček (University of Zagreb)

Abstract: Let $Z$ be an isotropic stable process in the Euclidean space. The process $Z$ is killed upon exiting an open set $D$ and the killed process is then subordinated by an independent $\gamma$-stable subordinator, $0<\gamma <1$. The resulting process is a Hunt process in $D$. In this talk, I will discuss several potential theoretical properties of this process such as Harnack inequality for nonnegative harmonic functions, the Carleson estimate, Green function and jumping kernel estimates in smooth sets $D$, and in particular, the boundary Harnack principle. Surprisingly, it turns out the BHP holds only if $1/2<\gamma<1$. This is joint work with Panki Kim and Renming Song.

Mean Field Analysis of Neural Networks in Machine Learning

Justin Sirignano (Illinois ISE)

Abstract: Neural network models in machine learning have revolutionized fields such as image, text, and speech recognition. There's also growing interest in using neural networks for applications in science, engineering, medicine, and finance. Despite their immense success in practice, there is limited mathematical understanding of neural networks. We mathematically study neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations. We rigorously prove that the neural network satisfies a Law of Large Numbers (LLN) and a Central Limit Theorem (CLT). The LLN is the solution of a nonlinear partial differential equation while the CLT satisfies a stochastic partial differential equation.

One-point function estimates and natural parametrization for loop-erased random walk in three dimensions

Xinyi Li (University of Chicago)

Abstract: In this talk, I will talk about loop-erased random walk (LERW) in three dimensions. I will first give an asymptotic estimate on the probability that 3D LERW passes a given point (commonly referred to as the one-point function). I will then talk about how to apply this estimate to show that 3D LERW as a curve converges to its scaling limit in natural parametrization. If time permits, I will also talk about the asymptotics of non-intersection probabilities of 3D LERW with simple random walk. This is a joint work with Daisuke Shiraishi (Kyoto).

On the structure of preferential attachment networks with community structure.

Bruce Hajek (ECE)

Abstract: An extensive theory of community detection has developed within the past few years. The goal is to discover clusters of vertices in a graph based on the edges of the graph. In this talk we focus on the problem of community detection for the Barabasi-Albert preferential attachment model with communities, defined by Jonathan Jordan. In such model, vertices are sequentially attached to the graph, with preference to attach more edges to existing vertices with larger degrees, multiplied by affinities based on community membership. It is shown that the model has sufficient structure to formulate approximate belief propagation algorithms for community detection. (Details at arxiv 1801.06818.) Based on joint work with S. Sankagiri.

Convergence of Trapezoid Rule to Rough Integrals

Zachary Selk (Purdue University)

Abstract: Rough paths offer the only notion of solution to SDEs driven by non-semimartingales with poor total $p$-variation such as fractional Brownian motion with Hurst index $H<1/2$. Rough paths further offer a notion of pathwise solution to SDEs classically handled by Itô calculus. Applications include large deviations theory, volatility models in finance, filtering theory, and SPDEs. If one can define certain iterated integrals, they act as a sort of "correction term" in Riemann sums, restoring convergence to classically divergent sums. These corrected Riemann sums are known as rough integrals. However these correction terms are unnatural. We prove for a general class of multidimensional Gaussian processes the convergence of the trapezoid rule to these corrected Riemann sums. The trapezoid rule doesn't have any correction terms so is in some sense more natural. Joint work with Yanghui Liu and Samy Tindel.