Local martingale functions of Brownian motion

Speaker: 

Professor Patrick Fitzsimmons

Institution: 

University of California, San Diego

Time: 

Tuesday, April 26, 2005 - 11:00am

Location: 

MSTB 254

ABSTRACT: A harmonic function of the Brownian path is a local martingale. Is the converse true? We show that the class of local martingale functions of Brownian motion is co-extensive with the class of finely harmonic functions, and then use a results of Fuglede and Gardiner to answer this question in the negative, in dimensions bigger than 2.

Renormalized self-intersection local time and the range of random walks.

Speaker: 

Professor Richard Bass

Institution: 

University of Connecticut

Time: 

Tuesday, February 15, 2005 - 11:00am

Location: 

MSTB 254

Self-intersection local time $\beta_t$ is a measure of how often
a Brownian motion (or other process) crosses itself. Since Brownian
motion in the plane intersects itself so often, a renormalization
is needed in order to get something finite. LeGall proved that
$E e^{\gamma \beta_1}$ is finite for small $\gamma$ and infinite
for large $\gamma$. It turns out that the critical value is related
to the best constant in a Gagliardo-Nirenberg inequality. I will discuss
this result (joint work with Xia Chen) as well as large deviations
for $\beta_1$ and $-\beta_1$ and LILs for $\beta_t$ and $-\beta_t$.
The range of random walks is closely related to self-intersection
local times, and I will also discuss joint work with Jay Rosen
making this idea precise.

Small ball probablities and the quantization problem for Gaussian measures.

Speaker: 

Professor Michael Scheutzow

Institution: 

Technische Universitat, Berlin

Time: 

Tuesday, March 1, 2005 - 11:00pm

Location: 

MSTB 254

Let $\mu$ be a probability measure on a metric space $(E,d)$ and $N$ a positive integer.
The {\em quantization error} $e_N$ of $\mu$ is defined as the infimum over all subsets ${\cal{E}}$
of $E$ of cardinality $N$ of the average distance w.~r.~t.~$\mu$ to the closest point in the set
${\cal{E}}$. We study the asymptotics of $e_N$ for large $N$. We concentrate on the
case of a Gaussian measure $\mu$ on a Banach space. The asymptotics of $e_N$ is closely related to
{\em small ball probabilities} which have received considerable interest in the past decade.
The quantization problem is motivated by the problem of encoding a continuous signal
by a specified number of bits with minimal distortion. This is joint work with Steffen Dereich,
Franz Fehringer, Anis Matoussi and Michail Lifschitz.

Random walks along orbits of dynamical systems'

Speaker: 

Associate Professor Vadim Kaloshin

Institution: 

Cal Tech

Time: 

Tuesday, May 17, 2005 - 11:00am

Location: 

MSTB 254

Consider a compact manifold $M$ (e.g. a torus) equipped with
a smooth measure $\mu$ (e.g. Lebesgue measure in the case
of torus) as a probability space $(M,\mathcal M,\mu)$. Consider
an ergodic map $T:M \to M$ along with a smooth function
$p:M \to (0,1)$. Define a random walk along orbits of $T$ as follows:
a point $x$ jumps to $T x$ with probability $p(x)$ and
to $T^{-1} x$ with probability $1-p(x)$.
Is there a limiting distribution of such a random walk for a generic
initial point? Is it absolutely continuous with respect to $\mu$?
We shall present an answer for several essentially different
maps $T$.

A tractable complex network model

Speaker: 

Professor David Aldous

Institution: 

University of California, Berkeley

Time: 

Tuesday, April 12, 2005 - 11:00am

Location: 

MSTB 254

We describe a stochastic model for complex networks possessing three
qualitative features: power-law degree distributions, local clustering, and
slowly-growing diameter.
The model is mathematically natural, permits a wide variety
of explicit calculations, has the desired three qualitative features,
and fits the complete range of degree scaling exponents and clustering parameters.
Write-ups exist as a

short version
and as a
long version

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