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

Vassili N. Kolokoltsov, University of Warwick, UK.

¿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.