In recent years, there has been an upsurge of interest in using techniques drawn from probability to tackle problems in analysis. These applications arise in subjects such as potential theory, harmonic analysis, singular integrals, and the study of analytic functions. This book presents a modern survey of these methods at the level of a beginning Ph.D. student. Highlights of this book include the construction of the Martin boundary, probabilistic proofs of the boundary Harnack principle, Dahlberg's theorem, a probabilistic proof of Riesz' theorem on the Hilbert transform, and Makarov's theorems on the support of harmonic measure. The author assumes that a reader has some background in basic real analysis, but the book includes proofs of all the results from probability theory and advanced analysis required. Each chapter concludes with exercises ranging from the routine to the difficult. In addition, there are included discussions of open problems and further avenues of research.

Probabilistic Techniques in Analysis (Probability and Its Applications)

Science & Math

Bass, Richard F.

Mathematics

Springer

Diffusions and Elliptic Operators (Probability and Its Applications)

A discussion of the interplay of diffusion processes and partial differential equations with an emphasis on probabilistic methods. It begins with stochastic differential equations, the probabilistic machinery needed to study PDE, and moves on to probabilistic representations of solutions for PDE, regularity of solutions and one dimensional diffusions. The author discusses in depth two main types of second order linear differential operators: non-divergence operators and divergence operators, including topics such as the Harnack inequality of Krylov-Safonov for non-divergence operators and heat kernel estimates for divergence form operators, as well as Martingale problems and the Malliavin calculus. While serving as a textbook for a graduate course on diffusion theory with applications to PDE, this will also be a valuable reference to researchers in probability who are interested in PDE, as well as for analysts interested in probabilistic methods.

This comprehensive guide to stochastic processes gives a complete overview of the theory and addresses the most important applications. Pitched at a level accessible to beginning graduate students and researchers from applied disciplines, it is both a course book and a rich resource for individual readers. Subjects covered include Brownian motion, stochastic calculus, stochastic differential equations, Markov processes, weak convergence of processes and semigroup theory. Applications include the Black-Scholes formula for the pricing of derivatives in financial mathematics, the Kalman-Bucy filter used in the US space program and also theoretical applications to partial differential equations and analysis. Short, readable chapters aim for clarity rather than full generality. More than 350 exercises are included to help readers put their new-found knowledge to the test and to prepare them for tackling the research literature.

Stochastic Processes (Cambridge Series in Statistical and Probabilistic Mathematics)

Cambridge University Press

A long open problem in probability theory has been the following: Can the graph of planar Brownian motion be split by a straight line? Let $Z_t$ be two-dimensional Brownian motion. Say that a straight line $\mathcal L$ is a cut line if there exists a time $t \in (0,1)$ such that the trace of $\{ Z_s: 0 \leq s \lt t\}$ lies on one side of $\mathcal L$ and the trace of $\{Z_s: t \lt s \lt 1\}$ lies on the other side of $\mathcal L$. In this volume, the authors provide a solution, discuss related works, and present a number of open problems.

Cutting Brownain Paths (Memoirs of the American Mathematical Society)

BURDZY, KRZYSZTOF

Amer Mathematical Society

Moderate Deviations for the Range of Planar Random Walks.

Moderate Deviations for the Range of Planar Random Walks (Memoirs of the American Mathematical Society)

Rosen, Jay

Chen, Xia

This comprehensive guide to stochastic processes gives a complete overview of the theory and addresses the most important applications. Pitched at a level accessible to beginning graduate students and researchers from applied disciplines, it is both a course book and a rich resource for individual readers. Subjects covered include Brownian motion, stochastic calculus, stochastic differential equations, Markov processes, weak convergence of processes and semigroup theory. Applications include the Black-Scholes formula for the pricing of derivatives in financial mathematics, the Kalman-Bucy filter used in the US space program and also theoretical applications to partial differential equations and analysis. Short, readable chapters aim for clarity rather than full generality. More than 350 exercises are included to help readers put their new-found knowledge to the test and to prepare them for tackling the research literature.

Stochastic Processes (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 33)

Nearly every Ph.D. student in mathematics needs to take a preliminary or qualifying examination in real analysis. This book provides the necessary tools to pass such an examination.Clarity: Every effort was made to made to present the material in as clear a fashion as possible.Lots of exercises: Over 220 exercises, ranging from routine to challenging, are presented. Many are taken from preliminary examinations given at major universities.Affordability: The book is priced at well under $20.

Real analysis for graduate students: measure and integration theory

CreateSpace Independent Publishing Platform

Diffusions and Elliptic Operators

This book is a course on real analysis (measure and integration theory plus additional topics) designed for beginning graduate students. Its focus is on helping the student pass a preliminary or qualifying examination for the Ph.D. degree.

Real Analysis for Graduate Students, Second Edition

Nearly every Ph.D. student in mathematics needs to take a preliminary or qualifying examination in real analysis. This book provides the necessary tools to pass such an examination. 
<ul>
<li> Clarity: Every effort was made to present the material in as clear a fashion as possible. </li>
<li> Lots of exercises: Over 400 exercises, ranging from routine to challenging, are presented. Many are taken from preliminary examinations given at major universities. </li>
<li> Affordability: This book is priced at well under $15. </li>
</ul>
 Topics covered include the definition and construction of measures, the Lebesgue integral and its properties, limit theorems, product measures, signed measures, the Radon-Nikoydym theorem, differentiation, Lp spaces, Fourier transforms, the Riesz representation theorem, and Banach and Hilbert spaces.

Real Analysis for Graduate Students, version 2.1

See the best price to sell, buy, or rent books by Bass, Richard F.. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Books by Bass, Richard F.