Probability

Summer School in Probablility 2008

The School schedule ran 4 days per week to give participants ample time for study, interaction with other students and discovering Vancouver and its surroundings. Many explored the old growth forests at Lighthouse Park and Lynn Canyon and Headwaters Parks on the North Shore. Those who enjoy more strenuous hiking discovered the beauty of the surrounding mountains and ocean on a number of organized hikes. This year we went to the top of Anvil Island which is only accessible by water taxi from Horseshoe Bay. The 2500 ft.

Date: 
Wed, Jun 11, 2008
Class: 
Scientific
Subject: 
Mathematics
Probability

Random Walk in Random Scenery

Author: 
Frank den Hollander
Date: 
Fri, Feb 3, 2006
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 

In this talk we consider a random walk on a randomly colored lattice and ask what are the properties of the sequence of colors encountered by the walk.

Notes: 
PDF icon denhollander_3feb.pdf
Class: 
Scientific
Subject: 
Probability

Phase Transitions for Interacting Diffusions

Author: 
Frank den Hollander
Date: 
Wed, Jan 18, 2006
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 

In the present talk we focus on the ergodic behavior of systems of interacting diffusions.

Notes: 
PDF icon denhollander_18jan.pdf
Class: 
Scientific
Subject: 
Dynamical Systems
Probability

Algebraic Z^d-actions

Author: 
Klaus Schmidt
Date: 
Fri, Nov 1, 2002
Location: 
University of Victoria, Victoria, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 
This is a written account of five Pacific Institute for the Mathematical Sciences Distinguished Chair Lectures given at the Mathematics Department, University of Victoria, BC, in November 2002. The lectures were devoted to the ergodic theory of $\mathbb Z^d$--actions, i.e. of several commuting automorphisms of a probability space. After some introductory remarks on more general $\mathbb Z^d$-actions the lectures focused on ‘algebraic’ $\mathbb Z^d$-actions, their sometimes surprising properties, and their deep connections with algebra and arithmetic. Special emphasis was given to some of the very recent developments in this area, such as higher order mixing behaviour and rigidity phenomena.
Notes: 
PDF icon Schmidt.pdf
Class: 
Scientific
Subject: 
Group Theory
Probability

Entropy and Orbit Equivalence

Author: 
Daniel J. Rudolph
Date: 
Fri, Oct 1, 2004
Location: 
University of Victoria, Victoria, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 

In these notes we first offer an overview of two core areas in the dynamics of probability measure preserving systems, the Kolmogorov-Sinai theory of entropy and the theory of orbit equivalence. Entropy is a nontrivial invariant that, said simply, measures the exponential growth rate of the number of orbits in a dynamical system, a very rough measure of the complexity of the orbit structure. On the other hand, the core theorem of the orbit theory of these systems, due to Henry Dye, says that any two free and ergodic systems are orbit equivalent, that is to say can be regarded as sitting on the same set of orbits. The goal we set out to reach now is to explain and understand the seeming conflict between these two notions.

Notes: 
PDF icon Rudolph.pdf
Class: 
Scientific
Subject: 
Dynamical Systems
Probability

Self-Interacting Walk and Functional Integration

Author: 
David Brydges
Date: 
Thu, Sep 14, 2000
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 
These lectures are directed at analysts who are interested in learning some of the standard tools of theoretical physics, including functional integrals, the Feynman expansion, supersymmetry and the Renormalization Group. These lectures are centered on the problem of determining the asymptotics of the end-to-end distance of a self-avoiding walk on a $D$-dimensional simple cubic lattice as the number of steps grows. When $D=4$ the end-to-end distance has been conjectured to grow as Const. $n^{1/2}\log^{1/8}n,$ where $n$ is the number of steps. We include a theorem, obtained in joint work with John Imbrie, that validates the $D=4$ conjecture in the simplified setting known as the ``Hierarchial Lattice.''
Notes: 
PDF icon Brydges.pdf
Class: 
Scientific
Subject: 
Mathematical Physics
Probability

PDE from a probability point of view

Author: 
Richard Bass
Date: 
Thu, Jan 1, 2004
Location: 
UBC, Vancouver, Canada
Abstract: 

These notes are for a course I gave while on sabbatical at UBC. The topics covered are: stochastic differential equations, solving PDEs using probability, Harnack inequalities for nondivergence form elliptic operators, martingale problems, and divergence form elliptic operators.

This course presupposes the reader is familiar with stochastic calculus; see the notes on my web page for Stochastic Calculus, for example. These notes for the most part are based on my book Diffusions and Elliptic Operators, Springer-Verlag, 1997.

Notes: 
PDF icon Bass.pdf
Class: 
Scientific
Subject: 
Partial Differential Equations
Probability

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