Continuous Parameter Markov Processes and Stochastic Differential Equations

Beschreibung

This graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications.  The authors focus on developing context and intuition before formalizing the theory of each topic, illustrated with examples.
After a review of some background material, the reader is introduced to semigroup theory, including the Hille–Yosida Theorem,  used to construct continuous parameter Markov processes.  Illustrated with examples, it is a cornerstone of Feller’s seminal theory of the most general one-dimensional diffusions studied in a later chapter. This is followed by two chapters with probabilistic constructions of jump Markov processes,  and   processes with independent increments, or Lévy processes. The greater part of the book is devoted to  Itô’s fascinating theory of stochastic differential equations,  and to the study of  asymptotic properties of diffusions  in all dimensions, such as  explosion, transience, recurrence,  existence of steady states, and the speed of convergence to equilibrium.  A broadly applicable functional central limit theorem for ergodic Markov processes is presented with important examples. Intimate connections between diffusions  and linear second order elliptic and parabolic partial differential equations are laid out in two chapters, and are used for computational purposes.  Among Special Topics chapters, two study anomalous diffusions: one on  skew Brownian motion, and the other on an intriguing multi-phase homogenization of solute transport in porous media.

This graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications.  The authors focus on developing context and intuition before formalizing the theory of each topic, illustrated with examples.

After a review of some background material, the reader is introduced to semigroup theory, including the Hille–Yosida Theorem,  used to construct continuous parameter Markov processes.  Illustrated with examples, it is a cornerstone of Feller’s seminal theory of the most general one-dimensional diffusions studied in a later chapter. This is followed by two chapters with probabilistic constructions of jump Markov processes,  and   processes with independent increments, or Lévy processes. The greater part of the book is devoted to  Itô’s fascinating theory of stochastic differential equations,  and to the study of  asymptotic properties of diffusions  in all dimensions, such as   explosion, transience, recurrence,  existence of steady states,  and the speed of convergence to equilibrium.  A broadly applicable functional central limit theorem for ergodic Markov processes is presented with important examples. Intimate connections between diffusions  and linear second order elliptic and parabolic partial differential equations are laid out in two chapters, and are used for computational purposes.  Among Special Topics chapters, two study anomalous diffusions: one on  skew Brownian motion, and the other on an intriguing multi-phase homogenization of solute transport in porous media.

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Rabi Bhattacharya

Themen in »Continuous Parameter Markov Processes and Stochastic Differential Equations«

Hille-Yoshida theorem Lévy processes Markov processes with jumps Markov property central limit theorem infinitely divisible distributions jump phenomena semigroups

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“This book is rich in content and logically rigorous, making it an excellent reference for studying Markov processes and stochastic differential equations. After reading it, it can give everyone a clearer and deeper understanding of this field, which is very beneficial for those who are engaged in or interested in researching in this field.” (Jiankang Liu, zbMATH 1555.60001, 2025)

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