Reference book for Brownian Motion

Question

I want to know about books for reading Brownian motion. I am aware of measure theoretic probability theory.

Answer 1

The monograph Brownian Motion - An Introduction to Stochastic Processes by René Schilling & Lothar Partzsch contains a lot of material on Brownian motion; I like it pretty much. It starts with the basics (e.g. that it is a Gaussian process, a Markov process, how to construct Brownian motion,...), discusses the connection to PDEs and (the more general) theory of Markov processes, presents several results on path properties (iterated law of logarithm, Strassen's law,...) and, moreover, there are also several chapters on stochastic integrals with respect to Brownian motion. There are full solutions to all exercises available on the web.

Answer 2

I don't know what your background is but the standard introductory graduate texts are Brownian Motion and Stochastic Calculus by Karatzas and Shreve, and Continuous Martingales and Brownian Motion by Revuz and Yor.

I'm also a big fan of Probability and Stochastics by Çinlar. This one includes a lot more measure/probability background if you don't already have it.