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Markov processes represent a universal model for a large variety of real life random evolutions. The wide flow of new ideas, tools, methods and applications constantly pours into the ever-growing stream of research on Markov processes that rapidly spreads over new fields of natural and social sciences, creating new streamlined logical paths to its turbulent boundary. Even if a given process is not Markov, it can be often inserted into a larger Markov one (Markovianization procedure) by including the key historic parameters into the state space.
This monograph gives a concise, but systematic and self-contained, exposition of the essentials of Markov processes, together with recent achievements, working from the "physical picture" - a formal pre-generator, and stressing the interplay between probabilistic (stochastic differential equations) and analytic (semigroups) tools.
The book will be useful to students and researchers. Part I can be used for a one-semester course on Brownian motion, Lévy and Markov processes, or on probabilistic methods for PDE. Part II mainly contains the author's research on Markov processes.
From the contents:
- Tools from Probability and Analysis
- Brownian motion
- Markov processes and martingales
- SDE, DE and martingale problems
- Processes in Euclidean spaces
- Processes in domains with a boundary
- Heat kernels for stable-like processes
- Continuous-time random walks and fractional dynamics
- Complex chains and Feynman integral
List of contents
Part I Brownian Motion, Markov Processes, Martingales.
1 Preliminaries in Probability and Analysis.
2 Browninan Motion I: Constructions.
3 Martingales and Markov Processes.
4 Browninan Motion II: Elements of Analysis.
Part II Basic Constructions of Markov Semigroups.
1 Analytic Constructions.
2 Probabilistic Constructions.
3 Heat Kernel Estimates.
4 Process in Cones and Bounded Domains.
Part III Extensions, Developments, Applications.
1 CTRW and Fractional Dynamics.
2 Complex Markov Chains and Feynman integral.
3 Controlled Processes.
4 Semiclassical Asymptotic.
5 Miscellany.
6 Bibliographical Comments.
About the author
Vassili N. Kolokoltsov, University of Warwick, UK.
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"[...] As a whole, the book is a valuable source of information both for specialists in the field and those who study the theory of stochastic processes."
Anatoly N. Kochubei, Zentralblatt für Mathematik
