Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

When a fair coin is flipped many times, the order of the imbalance between the witnessed numbers of heads and tails is the square-root of the number of coin flips, and the scaled random imbalance is asymptotically Gaussian. By tracking this imbalance in scaled time, Brownian motion is obtained. This class will explore several natural random systems that give rise to scaling limits that could be called cousins of Brownian motion: processes that have important differences from, but also distinct similarities with, the classical object. We will study: (1) power-law urn and voter models, which give rise to a fractional version of Brownian motion; (2) motion in a preferred direction in a disordered medium, in which traps formed by the environment detain the walker and disrupt its Brownian advance; and (3) random turn games, in which players make strategic decisions about resource allocation in games with many turns. The scaled trajectory of the counter on the game board is governed by an infinity version of Brownian motion, which moves in a more singular fashion than does the usual, spherically symmetric, Brownian motion. In probing several directions such as these, the class aims to offer modes of entry into research in them.

Repeat Rules