**Purdue Probability Seminar Spring 2020**

## Wednesdays 1:30-2:20 PM, Location: UNIV 103

**1/15**

**Phanuel Mariano**

**(University of New Haven**

**)**webpage

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**Title**: Can you hear the fundamental frequency of a drum using probability

**Abstract**: In Mark Kac's famous 1966 paper, he asked ``Can you hear the shape of a drum?'' The precise question being, if you heard the full list of overtones and frequencies while you were blindfolded, would you be able to tell the shape of the drumhead \(D\subset\mathbb{R}^{2}\) in some mathematical way? The problem I will primarily speak about is in regards to how the fundamental frequency of a drum and probability theory are related. This connection will be through an inequality involving the fundamental frequency of a drum with drumhead \(D\subset\mathbb{R}^{d}\) and the maximum expected lifetime of Brownian motion started inside a domain \(D\subset\mathbb{R}^{d}\). We improve on the constants of the known inequality and prove a new asymptotically sharp inequality involving the moments of the expected lifetime of Brownian motion. We discuss conjectures about the sharp inequality and present our partial results about the extremal domains and sharp constants. This talk is based on joint work with Rodrigo Bañuelos and Jing Wang.

**1/22 Jonathan Peterson**

**(Purdue)**webpage

**Title**: Convergence of excited random walks on Markovian cookie stacks to Brownian motion perturbed at extrema

**Abstract**: Excited random walks are a class of self-interacting random walks where the transition probabilities depend on the local time of the walk at the present site. It has previously been shown, only for certain special cases, that when the walk is recurrent the limiting distribution of the rescaled path of the walk is a Brownian motion perturbed at its extrema. We extend this convergence to a much more general class of excited random walks: excited random walks on Markovian cookie stacks. For this we develop a completely new approach to proving convergence to perturbed Brownian motion. Previously, it was already know that generalized Ray-Knight Theorems for the excited random walk were consistent with convergence to perturbed Brownian motion. We use improved versions of these generalized Ray-Knight Theorems to construct a coupling of the random walk with a perturbed Brownian motion. This talk is based on joint work with Elena Kosygina and Tom Mountford.

**1/29**

**Samy Tindel (Purdue)**webpage

**Title**: The parabolic Anderson model in a rough environment

**Abstract**: The parabolic Anderson model has to be interpreted as a heat equation in a random environment. In this talk I will first recall some basic physical and mathematical facts about this model. In particular, I will describe the so-called localization phenomenon for eigenvectors.

In the second part of my talk I will focus on the description of the noisy Gaussian environment I’m interested in, especially in very rough (or singular) situations. I will then summarize the asymptotic behavior of moments obtained in the literature, including very recent results about critical cases.

If time allows it I will also give some hints about Feynman-Kac representations for the model, which are one of the the basic tools for its analysis.

**2/5 Christopher Gartland**

**(UIUC)**webpage

**Title**: Markov Chains in metric spaces

**Abstract**: The Ribe program is the research program concerned with generalizing local linear properties of Banach spaces to nonlinear, bi-Lipschitz invariant properties of metric spaces. We'll review local linear properties of Banach spaces and focus on those whose nonlinear generalizations involve finite Markov chains taking values in a metric space. Particular emphasis will be placed on an invariant called Markov convexity; we'll discuss results on the computation of this invariant for Carnot groups.

**2/12 Yiran Liu (Purdue)**

**Title**: \(G_t/G_t/\infty\) Queueing Model in Random Environment

**Abstract**: Nonstationary queueing models have been extensively studied in the literature on queues. In the Markovian settings, the arrival and service intensities are naturally assumed to be deterministic time-varying smooth functions. However, in practice, queueing systems are often subject to ``environmental" noise. In our study, we want to understand the stochastic fluctuations in the model intensities and its impact on system performance metrics. To this aim, we study a \(G_t/G_t/\infty\) infinite server queueing model imbedded in a random environment.

In this talk, we will briefly review the basics of queueing theory and Poisson point process. The description of our queueing model will be explained. I will present a new result on a homogenized process that can be approximated by an \(M_t/G_t/\infty\) queue with modified parameters.

This talk is based on joint work with Harsha Honnappa, Samy Tindel, and Aaron Yip.

**2/19 Mickey Salins**

**(Boston University)**webpage

**Title:**Existence and Uniqueness for the mild solution of the stochastic heat equation with non-Lipschitz drift on an unbounded spatial domain

**Abstract:**I prove the existence and uniqueness of the mild solution for a nonlinear stochastic heat equation defined on an unbounded spatial domain. The nonlinearity is not assumed to be globally, or even locally, Lipschitz continuous. Instead the nonlinearity is assumed to satisfy a one-sided Lipschitz condition. First, I introduce a strengthened version of the Kolmogorov continuity theorem to prove that the stochastic convolutions of the fundamental solution of the heat equation and a spatially homogeneous noise grow no faster than polynomially. Second, a deterministic mapping that maps the stochastic convolution to the solution of the stochastic heat equation is proven to be Lipschitz continuous on polynomially weighted spaces of continuous functions. These two ingredients enable the formulation of a Picard iteration scheme to prove the existence and uniqueness of the mild solution.

**2/26 Yongjia Xie (Purdue)**

**Title:**Functional weak limit of random walks in cooling random environment

**Abstract:**In the last few decades, many results of random walks in dynamic random environment have been made. One of the dynamics of the environment is called the cooling random environment which lets the underlying environment be refreshed on a deterministic time sequence. This model was first introduced by Luca Avena and Frank den Hollander in 2017. Later, they worked with Yuki Chino and Conrado da Costa to prove several properties such as LLN and CLT in different cooling regimes. In this talk, I will first introduce the basic setup and the above existing results of random walks in cooling random environment(RWCRE). Then I'll show my recent work about the functional weak limit of RWCRE under two cooling cases, namely, polynomial and exponential.

**3/4**

**3/11**

**Graham White (IU-Bloomington)**webpage

**Title:**A strong stationary time for the random transportation shuffle

**Abstract:**A natural problem in the study of Markov chains is how long it takes for a Markov chain of interest to mix. For the random walk on the symmetric group S_n, generated by transpositions, the mixing time in total variation distance is known to be precisely 1/2*nlog(n) - essentially, this is the time required to move each card at least once. The same result has been believed to be true for the separation distance mixing time, but the proof of the accepted upper bound contains a subtle error. I will discuss this random walk and give an alternate proof of the upper bound, re-establishing the result, using the combinatorial technique of strong stationary times. In a sense, this technique allows us to track `randomness' as it `builds up' as more and more cards are chosen.

**3/18 Spring Break**

**3/25**

**Swee hong Chan**

**(UCLA)**webpage

**4/1**

**4/3 (special seminar)**

**Lingjiong Zhu**

**(Florida State University)**webpage

**Title:**Stochastic Gradient Hamiltonian Monte Carlo for Non-Convex Stochastic Optimization

**Abstract:**Stochastic gradient Hamiltonian Monte Carlo (SGHMC) is a variant of stochastic gradient with momentum where a controlled and properly scaled Gaussian noise is added to the stochastic gradients to steer the iterates towards a global minimum. Many works reported its empirical success in practice for solving stochastic non-convex optimization problems, in particular it has been observed to outperform overdamped Langevin Monte Carlo-based methods such as stochastic gradient Langevin dynamics (SGLD) in many applications. Although asymptotic global convergence properties of SGHMC are well known, its finite-time performance is not well-understood. In this work, we study two variants of SGHMC based on two alternative discretizations of the underdamped Langevin diffusion. We provide finite-time performance bounds for the global convergence of both SGHMC variants for solving stochastic non-convex optimization problems with explicit constants. Our results lead to non-asymptotic guarantees for both population and empirical risk minimization problems. For a fixed target accuracy level, on a class of non-convex problems, we obtain complexity bounds for SGHMC that can be tighter than those for SGLD. These results show that acceleration with momentum is possible in the context of global non-convex optimization. This is based on the joint work with Xuefeng Gao and Mert Gurbuzbalaban.

**4/8**

**Xiaoming Song**

**(Drexel)**webpage

**4/15**

**Pierre Yves Gaudreau Lamarre**

**(Princeton)**webpage

**4/22**

**Aaron Nung Kwan Yip**

**(Purdue)**webpage

**4/24**

**(special seminar)**

**Adam Osekowski (**

**University of Warsaw)**webpage

**4/29**

**Elizabeth Meckes**

**(Case Western Reserve University)**webpage

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