Fall 2017

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Spring 2017

Trajectorial martingale transforms: Convergence and integration.

March 7

Convergence of the empirical spectral distribution of Gaussian matrix processes

March 15

For a given Gaussian symmetric matrix process ${Y}^{(n)}=\{{Y}^{(n)}(t){\}}_{t\ge 0}$ and a random symmetric matrix ${A}^{(n)}$ , independent of ${Y}^{(n)}$ , consider the process eigenvalues

$({\lambda}_{1}^{(n)}(t),\dots ,{\lambda}_{n}^{(n)}(t))$ associated to the matrix ${A}^{(n)}+{Y}^{(n)}(t)$ , as well as its corresponding process of empirical spectral measures

$${\mu}_{t}^{(n)}=\frac{1}{n}\sum _{j=1}^{n}{\delta}_{{\lambda}_{j}^{(n)}(t)}.$$

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

Abstract

April 19

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

This is a joint work with Jinho Baik.

November 30

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.

Admissible vector fields and quasi-invariant measures on path space

Let $x$ be a diffusion process taking values in a closed compact finite-dimensional manifold $M$ .

We discuss the construction of a class of vector fields $Z$ admissible with respect to the law of $x$ , and a quasi-invariance theorem for the flow of $Z$ . This talk is a continuation and the conclusion of my previous talk (September 28).

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 $|\rho (u)|\ge c$ .

**Denis Bell, **University of North Florida*
Divergence operators, transformations of measure, and
the interpolation method
*September 28

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 $T$ of the underlying measure space into a homotopy that interpolates between $T$ 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

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 $p>2$
and using techniques from Malliavin calculus.

February 3

In this talk, we consider the SDE model $d{X}_{t}=-\theta {X}_{t}dt+{\sigma}_{t}d{B}_{t}^{H}$ . Our goal is to estimate the ``integrated volatility'' ${\int}_{0}^{t}|{\sigma}_{s}{|}^{p}ds$ for an arbitrary $p>0$ , 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 ${X}_{t}$ . 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.

February 8

Let ${X}_{n}=\sum _{i=0}^{\mathrm{\infty}}{a}_{i}{\epsilon}_{n-i}$ , where the ${\epsilon}_{i}$ are i.i.d. centered random variables and $\sum _{i=0}^{\mathrm{\infty}}|{a}_{i}|<\mathrm{\infty}$ . Assume that $f$ is the probability density function of ${X}_{n}$ . We consider the estimation of the quadratic functional ${\int}_{R}{f}^{2}(x)dx$ . It is shown that, under certain mild conditions, the estimator

$$\frac{2}{n(n-1){h}_{n}}\sum _{1\le i<j\le n}K\left(\frac{{X}_{i}-{X}_{j}}{{h}_{n}}\right)$$

is asymptotically efficient.

Jason Siefken,

abstract

March 30

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.

November 11

We consider the parabolic Anderson model on ${\mathbb{Z}}^{d}$ 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 $(0,1)$ .

We then turn to the asymptotic time evolution of this solution. We will see that for $H\le 1/2$ , almost surely, it converges asymptotically to ${e}^{\lambda t}$ for some deterministic strictly positive constant $\lambda $ .

For $H>1/2$ on one hand, we demonstrate that the solution grows asymptotically no faster than ${e}^{kt\sqrt{\mathrm{log}t}}$ , for some positive deterministic constant $k$ . On the other hand, the asymptotic growth is lower-bounded by ${e}^{ct}$ for some positive deterministic constant $c$ .

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 ${\mathrm{\Phi}}_{3}^{4}$ 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.

November 19

Colloquium

November 25

December 2

We consider a sequence of processes, depending on a parameter $\theta \in (0,2\pi )$ , 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 $d$ -dimensional standard Brownian motion for any $d\ge 1$ . Using this procedure we can simulate easily independent Brownian motions.

December 9

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 $2/3<H<1$ . We analyze as well the limit behavior of a suitable renormalization of its $q$ -th chaotic component in the case $2/3<H<(4q-3)/(4q-2)$ . Our argument relies on an analysis of its chaotic components, based on Stein method and Malliavin calculus.

December 9, 4PM

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.

December 11, 1PM, Snow 302