Probability Theory Stochastic Processes with Applications

Chapter 1-2 of this text covers material of a basic probability course. Chapter 3 deals with discrete stochastic processes including Martingale theory. Chapter 4 covers continous time stochastic processes like Brownian motion and stochastic differential equations. The last chapter selected topics got considerably extended in the summer of 2006. In the original course, only localization and percolation problems were included. Now, more topic like estimation theory, Vlasov dynamics, multi-dimensional moment problems, random maps, circle-valued random variables, the geometry of numbers, Diophantine equations and harmonic analysis have been added. No previous knowledge in probability are necessary, but a basic exposure to calculus and linear algebra is required. Some real analysis as well as some background in topology, functional analysis and harmonic analysis can be helpful for the later chapters.

Oliver Knill grew up near Schaffhausen in Switzerland. He got his BA in Mathematics at ETH zurich with a senior theses in dynamical systems advised by Jurgen Moser. After a postgraduate visit at Technion in Haifa he wrote his PhD theses at the ETH zurich advised by Oscar Lanford III in the Area of dynamical systems and random operator theory. After three years in Pasadena at Caltech as a Ttaussky-Todd instructor in the group of Barry Simon, and one year at the University of Arizona in Tucson as a Hanno-Rund visiting assistant professor, he was a Swiss National Science foundation fellow at the University of Texas in Austin and worked a year as a research assistant at the Institute for Fusion Studies at UT. Since Fall 2000, he teaches at the Mathematics department at Harvard University. Oliver Knill started his research in the field of dynamical systems, tackling ergodic and spectral theoretical questions. He worked out and published Moser?s lecture notes? Selected Chapters in the calculus of variations? which appeared in 2003 in the series Lectures in Mathematics ETH Zurich More recently he works on applications of dynamical systems to probability theory and elementary number theory as well as inverse problem in analysis, geometry, and computer vision.