Description
Book SynopsisMarkov 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, ψDE 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
Table of ContentsPart 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.
