Markov processes, semigroups and generators.
The book consists of two parts. Part I is a well-written course on the theory of the Brownian motion and more general Markov processes touching, in more or less detail, many related subjects, such as martingales, Malliavin calculus, Lévy processes, various classes of stochastic integrals, stochastic differential equations and martingale problems. In Part II, the book becomes a research monograph, and the subjects covered here reflect the author's interests and achievements. In his words, ``the main results concern: {\parindent=7,5mm \begin{itemize}\item[(i)] various constructions and basic continuity properties of Markov processes, including processes stopped or killed at the boundary (as well as related boundary points classification and sensitivity analysis); \item[(ii)] in particular, heat kernel estimates for stable-like processes; \item[(iii)] limiting processes for position-dependent continuous time random walks (obtained by a random time change from the Markov processes) and related fractional (in time) dynamics; \item[(iv)] the rigorous Feynman path-integral representation for the solutions of the basic equations of quantum mechanics, via jump-type Markov processes. \end{itemize}} We also touch upon the theory of stochastic monotonicity, stochastic scattering, stochastic quasi-classical (also called small diffusion) asymptotics, and stochastic control.'' An interesting feature of the author's style is to give several alternative descriptions of the same object. Thus, he gives four different constructions of the Brownian motion (via approximation by piecewise-linear curves, via Hilbert space methods, via tightness of random-walk approximations, via the Kolmogorov-Chentsov continuity criterium), two constructions of the Poisson process, and several approaches to proving functional central limit theorems. The book contains an extensive bibliography. 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.
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