Brownian motion

Brownian Motion - By Peter Mörters and Yuval Peres

By Peter Mörters and Yuval Peres


Brownian motion is the single most important example of a stochastic process. The focus of this book is on sample path properties and the close connection to random walks- this connection is used both ways, e.g., the construction of Brownian motion and its local time are derived from
interpolation of random walks (following Paul Lévy), and conversely the Law of the iterated logarithm for random walks is derived by embedding them in Brownian motion. Viewing the Brownian path as a random fractal is another key theme of the book, and allows a detailed analysis of related sets of interest- zero sets, multiple points, cone points, fast and slow points. The connection with PDE (via Dirichlet’s problem) Stochastic integration and potential theory are explored as well. Classical notions in potential theory, such as regular points, are more naturally defined via Brownian motion than via the original analytic definition.

The direct approach of the book, with the only prerequisite being a solid background in measure theory, has led to its use for graduate courses in many universities.