A Criterion for Purely Absolutely Continuous Measures on Homogeneous Sets.

Speaker: 

Maxim Zinchenko

Institution: 

Caltech

Time: 

Thursday, January 22, 2009 - 2:00pm

Location: 

RH 306

Let $E$ be a homogeneous subset of $\mathbb{R}$ in the
sense of Carleson. Let $\mu$ be a finite positive measure on
$\mathbb{R}$ and $H_\mu(x)$ its Hilbert transform. We prove that
$\lim_{t\to\infty} t|E\cap\{x : |H_\mu(x)|>t\}|=0$ if and only if
$\mu_s(E)=0$, where $\mu_s$ is the singular part of $\mu$.

Heat kernel estimates for Dirichlet fractional Laplacian

Speaker: 

Professor Panki Kim

Institution: 

Seoul National University

Time: 

Tuesday, January 20, 2009 - 11:00am

Location: 

RH 306

In this talk, we discuss the sharp two-sided estimates for the heat kernel of Dirichlet fractional Laplacian in open sets. This heat kernel is also the transition density of a rotationally symmetric -stable process killed upon leaving an open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets. This is a joint work with Zhen-Qing Chen and Renming Song.

On the almost-sure invariance principle for random walk in random environment

Speaker: 

Professor Eiras Rassoul-Agha

Institution: 

University of Utah

Time: 

Tuesday, January 13, 2009 - 11:00am

Location: 

RH 306

Consider a crystal formed of two types of atoms placed at the nodes of the
integer lattice. The type of each atom is chosen at random, but the crystal
is statistically shift-invariant. Consider next an electron hopping from atom
to atom. This electron performs a random walk on the integer lattice with
randomly chosen transition probabilities (since the configuration seen by
the electron is different at each lattice site). This process is highly
non-Markovian, due to the interaction between the walk and the
environment.

We will present a martingale approach to proving the invariance principle
(i.e. Gaussian fluctuations from the mean) for (irreversible) Markov chains
and show how this can be transferred to a result for the above process
(called random walk in random environment).

This is joint work with Timo Sepp\"al\"ainen.

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