# 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.

## Random Walk in Random Scenery

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.

## Phase Transitions for Interacting Diffusions

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

## Algebraic Z^d-actions

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## Entropy and Orbit Equivalence

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.

## Self-Interacting Walk and Functional Integration

## PDE from a probability point of view

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.