Department of Mathematics

Probability and Statistics Seminar

Fall 2017
Wednesdays, 4:00 p.m., 306 Snow Hall
Please send Terry or Zhipeng an email if you are interested in receiving email notifications.

Spring 2017

Trajectorial martingale transforms: Convergence and integration.
February 22
Abstract:
Starting with a trajectory space, providing a non-stochastic analogue of a martingale process, we use the notion of super-replication to introduce a definition for a null function and the associated notion of a property holding almost everywhere (a.e.). The latter providing what can be seen as the worst case analogue of sets of measure zero in a stochastic setting. The a.e. notion is used to prove the pointwise convergence, on a full set of the original trajectory space, of the limit of a trajectorial martingale transform sequence. The setting also allows to construct a natural integration operator.

Yanhao Cui
Preliminary examination, SNOW 456.
March 7

Arturo Jaramillo Gill
Convergence of the empirical spectral distribution of Gaussian matrix processes

March 15
Abstract:
For a given Gaussian symmetric matrix process ${Y}^{\left(n\right)}=\left\{{Y}^{\left(n\right)}\left(t\right){\right\}}_{t\ge 0}$ $Y^{(n)}=\{Y^{(n)}(t)\}_{t\geq0}$ and a random symmetric matrix ${A}^{\left(n\right)}$ $A^{(n)}$, independent of ${Y}^{\left(n\right)}$ $Y^{(n)}$, consider the process eigenvalues
$\left({\lambda }_{1}^{\left(n\right)}\left(t\right),\dots ,{\lambda }_{n}^{\left(n\right)}\left(t\right)\right)$ $(\lambda_{1}^{(n)}(t),\dots, \lambda_{n}^{(n)}(t))$ associated to the matrix ${A}^{\left(n\right)}+{Y}^{\left(n\right)}\left(t\right)$ $A^{(n)}+Y^{(n)}(t)$, as well as its corresponding process of empirical spectral measures
${\mu }_{t}^{\left(n\right)}=\frac{1}{n}\sum _{j=1}^{n}{\delta }_{{\lambda }_{j}^{\left(n\right)}\left(t\right)}.$

We prove that, under some mild conditions on ${Y}^{\left(n\right)}$ $Y^{(n)}$ and ${A}^{\left(n\right)}$ $A^{(n)}$,  the process $\left\{{\mu }_{t}^{\left(n\right)}{\right\}}_{t\ge 0}$ $\{\mu_{t}^{(n)}\}_{t\geq0}$ converges in probability to a deterministic limit $\left\{{\mu }_{t}{\right\}}_{t\ge 0}$ $\{\mu_{t}\}_{t\geq0}$, in the topology of weak convergence of probability measures, and characterize the Cauchy transform of ${\mu }_{t}$ $\mu_{t}$, in terms of the solution of a Burgers' equation.

Sayan Banerjee, UNC
Coupling, geometry and hypoellipticity
Abstract
March 29

Tiefeng Jiang, Minnesota
Random restricted partitions
April 19
Abstract:
We study two types of probability measures on the set of integer partitions of n  with at most m parts. The first one chooses the random partition with a chance related to its largest part only. We then obtain the limiting distributions of all of the parts together and that of the largest part as n tends to infinity while m is fixed or tends to infinity. In particular, if m goes to infinity not fast enough, the largest part satisfies the central limit theorem. The second measure is very general. It includes the Dirichlet distribution and the uniform distribution as special cases. We derive the asymptotic distributions of the parts jointly and that of the largest part by taking limit of n and m in the same manner as that in the first probability measure. For one case, the largest part has the Poisson-Dirichlet distribution asymptotically. This is a joint work with Ke Wang.

Note that Tiefeng Jiang will also be giving the departmental colloquium the next day.

Yiying Cheng
Generalized moment estimation for alpha-stable Ornstein-Uhlenbeck processes from discrete observations
April 21, 1 PM, SNOW 408.
Abstract:
This talk will be a part of Yiying's preliminary examination.

Zhipeng Liu, Courant
Multi-time, multi-space distribution of periodic TASEP
April 26
Abstract:
Determining the space and time process of the KPZ universal class is an outstanding problem. We consider this question for the periodic TASEP. First, we obtain a general formula for the multi-time, multi-space distribution for finite time assuming general initial condition. For periodic step initial condition, the formula is simplified to an integral of a Fredholm determinant. We then compute the limit in the large time limit in the so-called relaxation time scale, in which the spatial correlations are of the same order as the space period.

This is a joint work with Jinho Baik.

Fall 2016

Nicholas Ma
Rate of convergence for the weighted Hermite variations of the fractional Brownian motion
November 30
Abstract:
We obtain a sharp rate of convergence in the central limit theorem for higher order weighted variations of the fractional Brownian motion. The proof is based on the techniques of Malliavin calculus and the quantitative stable limit theorems proved by Nourdin, Nualart and Peccati.

This seminar talk will be apart of Nicholas Ma's preliminary examination.

Denis Bell, University of North Florida
Admissible vector fields and quasi-invariant measures on path space
October 26
Abstract:
Let $x$ $x$ be a diffusion process taking values in a closed compact finite-dimensional manifold $M$ $M$.
We discuss the construction of a class of vector fields $Z$ $Z$ admissible with  respect to the law of $x$ $x$, and a quasi-invariance theorem for the flow of $Z$ $Z$. This talk is a continuation and the conclusion of my previous talk (September 28).

Le Chen
Regularity of densities for stochastic fractional heat equation with rough initial data
October 5.
Abstract:
In this talk, I will present a recent joint work with Yaozhong Hu and David Nualart on the regularity of densities for stochastic fractional heat equation with rough initial data. I will show how rough initial data brings many difficulties to the problem and explain the way that we handle these difficulties. Except the rough initial data, another contribution in this project is that we can get rid of the nondegenerate condition $|\rho \left(u\right)|\ge c$ $|\rho(u)|\ge c$.

Denis Bell, University of North Florida
Divergence operators, transformations of measure, and the interpolation method
September 28
Abstract:
Divergence operators in infinite-dimensional Gaussian spaces play a central role in the development of the Malliavin calculus and related theories. In a different direction, there exists an extensive body of work concerning the transformation of Gaussian measure under maps with different types and degrees of regularity. In this talk, we develop a relationship between these two lines of research. The basic idea is the embedding of a transformation $T$ $T$ of the underlying measure space into a homotopy that interpolates between $T$ $T$ and the identity map. We use this method to establish the quasi-invariance of the law of a diffusion process on a compact differentiable manifold under a class of suitably constructed vector fields.

Arturo Kohatsu-Higa, Ritsumeikan University, Japan
Regularity of the law for an exit time for one-dimensional elliptic diffusions
September 21
Abstract:
In this talk we consider for simplicity the exit time of a one dimensional uniformly elliptic diffusion and
discuss the regularity properties of the density of this exit time under two different regularity conditions on the coefficients.

This is joint work with Noufel Frikha (Universite Paris Diderot) and Libo Li (University of New South Wales)

Denis Bell, University of North Florida
Stochastic Delay Equations and the Malliavin Calculus

September 7
Abstract

David Nualart

A new method for proving tightness based on Malliavin calculus.

August 31

Abstract:
We will show a functional central limit theorem for the renormalized self-intersection local time of the fractional Brownian motion. The tightness of the sequence of laws is derived by estimating moments of order $p>2$ $p>2$ and using techniques from Malliavin calculus.

Fall 2015/Spring 2016

Hong-Juan Zhou
Asymptotic Behavior of Power Variation of an Integral Fractional Process with Application to Parameter Estimation (Prelim-exam)
February 3
Abstract:
In this talk, we consider the SDE model  $d{X}_{t}=-\theta {X}_{t}dt+{\sigma }_{t}d{B}_{t}^{H}$ $dX_t = - \theta X_t dt + \sigma_t dB_t^H$.  Our goal is to estimate the integrated volatility'' ${\int }_{0}^{t}|{\sigma }_{s}{|}^{p}ds$ $\int_0^t | \sigma_s | ^p ds$ for an arbitrary $p>0$ $p > 0$, which is of great interest in many fields, such as Finance and Physics.  The estimator is generated based on the power variation of the observation sequence of ${X}_{t}$ $X_t$.  The asymptotic behavior of the power variation using high order differences will be presented using Malliavin Calculus.  Furthermore, by calculating the asymptotic variance of the estimator, a scenario on choosing the best order of difference is provided.

Fangjun Xu, East China Normal University
Stochastic Adaptive Control - Interdisciplinary Research Seminar
Kernel entropy estimation for linear processes
February 8
Abstract:
Let ${X}_{n}=\sum _{i=0}^{\mathrm{\infty }}{a}_{i}{\epsilon }_{n-i}$ $X_n=\sum\limits_{i=0}^\infty a_i \varepsilon_{n-i}$, where the ${\epsilon }_{i}$ $\varepsilon_i$ are i.i.d. centered random variables and $\sum _{i=0}^{\mathrm{\infty }}|{a}_{i}|<\mathrm{\infty }$ $\sum^{\infty}_{i=0}|a_i|<\infty$.  Assume that $f$ $f$ is the probability density function of ${X}_{n}$ $X_n$.  We consider the estimation of the quadratic functional ${\int }_{R}{f}^{2}\left(x\right)dx$ $\int_{R} f^2(x) dx$. It is shown that, under certain mild conditions, the estimator
$\frac{2}{n\left(n-1\right){h}_{n}}\sum _{1\le i

is asymptotically efficient.

March 23 (Double feature)

Jason Siefken,
Northwestern
Waiting time bounds for a subset of the Kari-Culik Tilings
Abstract:
A Wang tiling of the plane is a tiling of the plane by equal-sized squares with colored edges following the rule that two tiles may lie adjacent only if the colors on their shared edge match.  In the 1960's, it was discovered that there are sets of Wang tiles that only tile aperiodically.  The first aperiodic tile set had 20426 different tiles.  Since then this number has been reduced to 104 then to 56 and in 1996 a 13 piece tile set was found by Kari and Culik.  Whereas in all previous examples, aperiodicity could be explained using a hierarchical structure, the explanation of the Kari-Culik tiling's aperiodicity is number-theoretic.  This talk will give an overview of the Kari-Culik tilings as well as bounds on the waiting time to see particular patterns under horizontal and vertical translation.

Tempered Fractional Processes
abstract

David Herzog, Iowa State
On constructing optimal Lyapunov functions: an example.
March 30
Abstract: We discuss certain, explosive ODEs in the plane that become stable under the addition of noise. In each equation, the process by which stabilization occurs is intuitively clear: Noise diverts the solution away from any instabilities in the underlying ODE. However, in many cases, proving rigorously this phenomenon occurs has thus far been difficult and the current methods used to do so are rather ad hoc. Here we present a general, novel approach to showing stabilization by noise via the construction of Lyapunov functions and apply it to these examples. We will see that the methods used streamline existing arguments as well as produce optimal results, in the sense that they allow us to understand well the asymptotic behavior of the equilibrium measure at infinity.

David Nualart
Decomposition of Gaussian processes arising from spdes
September 30
Abstract:
We consider a class of self-similar Gaussian processes related to the solution of the stochastic heat equation driven by a colored noise. We establish a decomposition of such processes as the sum of a fractional Brownian motion plus a regular process and we discuss some applications of this decomposition.

Kamran Kalbasi, University of Warwick
Parabolic Anderson model driven by fractional noise
November 11
Abstract:
We consider the parabolic Anderson model on ${\mathbb{Z}}^{d}$ $\mathbb{Z}^d$ driven by fractional noise. We will see how we can prove that it has a mild solution given by Feynman-Kac representation . Our argument works in a unified way for every Hurst parameter in $\left(0,1\right)$ $(0,1)$.
We then turn to the asymptotic time evolution of this solution. We will see that for $H\le 1/2$ $H\leq1/2$, almost surely, it converges asymptotically to ${e}^{\lambda t}$ $e^{\lambda t}$ for some deterministic strictly positive constant $\lambda$ $\lambda$.
For $H>1/2$ $H>1/2$ on one hand, we demonstrate that the solution grows asymptotically no faster than ${e}^{kt\sqrt{\mathrm{log}t}}$ $e^{k t\sqrt{\log t}}$, for some positive deterministic constant $k$ $k$. On the other hand, the asymptotic growth is lower-bounded by ${e}^{ct}$ $e^{c t}$ for some positive deterministic constant $c$ $c$.

Weijun Xu, University of Warwick
Large scale behaviour of phase coexistence models
November 18
Abstract
The solutions to many singular SPDEs are obtained as limits of regularised and then renormalised equations. The renormalisation changes the original'' equation via quantities that are typically infinity, but they do have concrete physical meanings. As an example, I will explain how the ${\mathrm{\Phi }}_{3}^{4}$ $\Phi^4_3$ equation, interpreted after suitable renormalisations, arise naturally as the universal limit of symmetric phase coexistence models. We will also see how this universality can be lost when symmetry is broken.

Khoa Le, MSRI
November 19
Colloquium

Thanksgiving break
November 25

Giulia Binotto, La Universitat de Barcelona
Almost sure convergence to complex Brownian motion
December 2
Abstract:
We consider a sequence of processes, depending on a parameter $\theta \in \left(0,2\pi \right)$ $\theta\in(0,2\pi)$, constructed from a unique Poisson process and a family of independent Bernoulli random variables. We show that the sequence converges weakly to a complex Brownian motion and we build realizations of these processes that converge almost surely to a complex Brownian motion. We also derive a rate of convergence.
The results are based on a theorem of Skorohod. These results are extended in order to build a family of processes that converges almost surely to a $d$ $d$-dimensional standard Brownian motion for any $d\ge 1$ $d\ge1$. Using this procedure we can simulate easily independent Brownian motions.

Arturo Jaramillo Gil
Asymptotic properties of the derivative of the self-intersection local time of the fractional Brownian motion (Prelim-exam)
December 9, 3PM
Abstract
We consider an approximation of the derivative of the self-intersection local time of a fractional Brownian motion. We prove a central limit theorem for such approximation in the case  $2/3 $2/3. We analyze as well the limit behavior of a suitable renormalization of its $q$ $q$-th chaotic component in the case $2/3 $2/3 . Our argument relies on an analysis of its chaotic components, based on Stein method and Malliavin calculus.

Yanghui Liu
A Milstein-type scheme without Levy area terms for SDEs driven by fractional Brownian motion
December 9, 4PM
Abstract
We study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We review some recent results on an implementable scheme for these equations, which is based on a second-order Taylor expansion, where the usual Lévy area terms are replaced by products of increments of the driving fBm.

Yanghui Liu
Numerical Solution of Stochastic Differential Equations (Comp-exam)
December 11, 10:30AM, Snow 302

Chen Su
Sequential Implicit Sampling Methods for Bayesian Inverse Problems (Comp-exam)
December 11, 1PM, Snow 302