Edward C. Waymire - Böcker

"Probability and Partial Differential Equations in Modern Applied Mathematics" is devoted to the role of probabilistic methods in modern applied mathematics from the perspectives of both a tool for analysis and as a tool in modeling. There is a recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. There is also a growing appreciation of the role for the inclusion of stochastic effects in the modeling of complex systems. This has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations.This volume will be useful to researchers and graduate students interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in the sciences and engineering.

"Probability and Partial Differential Equations in Modern Applied Mathematics" is devoted to the role of probabilistic methods in modern applied mathematics from the perspectives of both a tool for analysis and as a tool in modeling. There is a recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. There is also a growing appreciation of the role for the inclusion of stochastic effects in the modeling of complex systems. This has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations.This volume will be useful to researchers and graduate students interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in the sciences and engineering.

This textbook offers an approachable introduction to stochastic processes that explores the four pillars of random walk, branching processes, Brownian motion, and martingales. Themes span Poisson processes, branching processes, the Kolmogorov–Chentsov theorem, martingales, renewal theory, and Brownian motion.

This textbook explores two distinct stochastic processes that evolve at random: weakly stationary processes and discrete parameter Markov processes.

This textbook explores two distinct stochastic processes that evolve at random: weakly stationary processes and discrete parameter Markov processes.

This graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications.

This graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications.

This volume presents some of the most influential papers published by Rabi N. Bhattacharya, along with commentaries from international experts, demonstrating his knowledge, insight, and influence in the field of probability and its applications.  For more than three decades, Bhattacharya has made significant contributions in areas ranging from theoretical statistics via analytical probability theory, Markov processes, and random dynamics to applied topics in statistics, economics, and geophysics.  Selected reprints of Bhattacharya’s papers are divided into three sections: Modes of Approximation, Large Times for Markov Processes, and Stochastic Foundations in Applied Sciences.  The accompanying articles by the contributing authors not only help to position his work in the context of other achievements, but also provide a unique assessment of the state of their individual fields, both historically and for the next generation of researchers.Rabi N. Bhattacharya: Selected Papers will be a valuable resource for young researchers entering the diverse areas of study to which Bhattacharya has contributed.  Established researchers will also appreciate this work as an account of both past and present developments and challenges for the future.

This text develops the necessary background in probability theory underlying diverse treatments of stochastic processes and their wide-ranging applications. In this second edition, the text has been reorganized for didactic purposes, new exercises have been added and basic theory has been expanded.  General Markov dependent sequences and their convergence to equilibrium is the subject of an entirely new chapter. The introduction of conditional expectation and conditional probability very early in the text maintains the pedagogic innovation of the first edition; conditional expectation is illustrated in detail in the context of an expanded treatment of martingales, the Markov property, and the strong Markov property. Weak convergence of probabilities on metric spaces and Brownian motion are two topics to highlight. A selection of large deviation and/or concentration inequalities ranging from those of  Chebyshev, Cramer–Chernoff, Bahadur–Rao, to Hoeffding have been added,with illustrative comparisons of their use in practice. This also includes a treatment of the Berry–Esseen error estimate in the central limit theorem.The authors assume mathematical maturity at a graduate level; otherwise the book is suitable for students with varying levels of background in analysis and measure theory. For the reader who needs refreshers, theorems from analysis and measure theory used in the main text are provided in comprehensive appendices, along with their proofs, for ease of reference.Rabi Bhattacharya is Professor of Mathematics at the University of Arizona. Edward Waymire is Professor of Mathematics at Oregon State University. Both authors have co-authored numerous books, including a series of four upcoming graduate textbooks in stochastic processes with applications.

Bhattacharya: Selected Papers will be a valuable resource for young researchers entering the diverse areas of study to which Bhattacharya has contributed.