**Purdue Probability Seminar Spring 2019**

**Date**

**Speaker****1/16**

**Hugo Panzo University of**

**Connecticut**

**Title**: A probabilistic approach to an inhomogeneous domain functional

**Abstract**: Inspired by recent research on the domain functional which maps a bounded Euclidean domain to the product of its first Dirichlet eigenvalue times the maximum of its torsion function, we study an inhomogenous version of this functional in dimension 1 and obtain upper and lower bounds using a probabilistic approach. The emphasis of the talk is on techniques and we introduce some probabilistic tools such as the Ray-Knight theorem, Khas'minskii's lemma, and the Boué-Dupuis-Borell formula.

**1/23 Zachary Selk Purdue University**

**Title**: Convergence of Trapezoid Rule to Rough Integral

**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.

**1/30**

**No talk due to extreme weather condition**

**2/6**

**Xinyi Li**

**University of Chicago**

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

**Abstract**: In this talk, I will talk about the 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 joint work with Daisuke Shiraishi (Kyoto).

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2/13 Changji Xu

2/13 Changji Xu

**University of Chicago**

**Title:**Random walk among Bernoulli obstacles

**Abstract:**Consider a discrete time simple random walk on Z d , d ≥ 2 with random Bernoulli obstacles, where the random walk will be killed when it hits an obstacle. We show that the following holds for a typical environment: conditioned on survival up to time n, the random walk will be localized in a single island. In addition, the limiting shape of the island is a ball and the asymptotic volume is also determined. This is based on joint works with Jian Ding. Time permitting, I will also describe a recent result in the annealed case, which is a joint work with Jian Ding, Ryoki Fukushima and Rongfeng Sun.

**2/20 Daesung Kim Purdue University**

**Title**: Quantitative inequalities for the expected lifetime of the Brownian motion

**Abstract:**The isoperimetric-type inequality for the expected lifetime of the Brownian motion state that the $L^p$ norm of the expected lifetime in a region is maximized when the region is a ball with the same volume. In particular, if $p=1$, it is called the Saint-Venant inequality and has a close relation to the classical Faber—Krahn inequality for the first eigenvalue. In this talk, we prove a quantitative improvement of the inequalities, which explains how a region is close to being a ball when equality almost holds in these inequalities. We also discuss some related open problems.

**2/27 Yanghui Liu Purdue University**

**Title**: Integrability for rough difference equations

**Abstract:**Integrability and sharp tail estimates is a key to density, ergodicity, and other related problems of a stochastic dynamical system. In the first part of the talk, we review the integrability results for Gaussian driven rough differential equations. Then in the second part, we consider the uniform integrability problem for difference equations, motivated by numerical approximation and its density estimate problems. We will focus on the new difficulties in this discrete model. The talk is based on a joint work with J. Le\'on, C. Ouyang, and S. Tindel.

**3/6**

**Tai Melcher**

**University of Virginia**

**Title:**Small-time asymptotics of subRiemannian Hermite functions"

**Abstract:**As in the Riemannian setting, a subRiemannian heat kernel is controlled by the geometry of the underlying manifold. In particular, the asymptotic behavior of the kernel can reveal certain geometric and topological data. We study the logarithmic derivatives of subRiemannian heat kernels in some cases and show that, under appropriate scaling, they converge to their analogues on stratified groups. This gives one quantification of the now standard idea that stratified groups play the role of the tangent space to subRiemannian manifolds. This is joint work with Joshua Campbell.

**3/13**

**Spring Break**

**3/20**

**Fabian Harang**University of Oslo**Title:**A multiparameter sewing lemma and applications

**Abstract:**We present an extension of the sewing lemma to multi parameter integrands. We apply this to study existence and uniquness of hyperbolic equations driven by Hölder fields on hypercubes in a Young regime. Furthermore, we discuss the challenges of extending the results to a "rough path" regime, and we will propose further applications.

**3/27**

**Gennady Samorodnitsky**

**Cornel University**

**Title:**High minima of Gaussian processes

**Abstract:**How likely is an entire piece of the sample path of a Gaussian process to lie above a high level? If this happens, how? What are the "critical points" in the sample path that assure this event? Are the answers to these questions different when the processes are smooth from the case when they are non-smooth? We will discuss these and related issues.

**4/3 Li Chen University of Connecticut**

**Title:**Gundy-Varopoulos martingale transforms and their projection operators

**Abstract:**We prove the $L^p$ boundedness of generalized first order Riesz transforms obtained as conditional expectations of martingale transforms \'a la Gundy-Varopoulos for quite general diffusions on manifolds and vector bundles. Several specific examples and applications will be presented. This is joint work with Rodrigo Banuelos and Fabrice Baudoin.

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4/10 Yimin Xiao Michigan State University

4/10 Yimin Xiao Michigan State University

**Title:**Some Local Properties of Parabolic Stochastic Partial Differential Equations

**Abstract:**Let \((t\,,x)\mapsto u_t(x)\) denote the solution to the stochastic PDE

\[\partial_tu=\mathscr{L} u_t(x)+\lambda\sigma(u_t(x) \dot{W}).\]

where the variable x ranges over \(\mathbb{R}, t>0\), \(\mathscr{L}\) denotes the generator of a symmetric \(\alpha\)-stable Levy process on \(\mathbb{R}\), and \(\dot{W}\) denotes space-time white noise on \((0,\infty)\times\mathbb{R}\). We prove a quantitative version of the intuitively-appealing statement that ``locally, \(t\mapsto u_t(x)\) behaves as a conditionally-Gaussian process.'' We then apply that statement in order to derive a number of detailed results about the local behavior of \(t\mapsto u_t(x)\), where \(x\in\mathbb{R}\) is fixed. Those results include facts such as iterated-logarithm-type behavior, and analysis of sample-function variations. This is a joint work with Davar Khoshnevisan, Jason Swanson, and Liang Zhang.

**4/17 Qing Yang Purdue University**

**Title:**Statistical Analysis of Spiked Tensor Models

**Abstract:**The goal of this paper is to provide a technical tool for making statistical inference on noisy tensor decomposition. We discuss the principal component analysis problem in a single-spike tensor model. Our major focus is its statistical properties, beyond the existing estimation algorithms. New theoretical insights on tensor PCA will be delivered.

**4/24 Atilla Yilmaz Temple University**

**Title:**Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential

**Abstract:**I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.

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