M.A. Lifshits – författare

This text presents a mathematically complete series of the most essential properties of Gaussian random functions. It focuses on a number of fundamental objects in the theory of Gaussian random functions and exposes their interrelations. The basic plots presented in the book embody: the kernel of a Gaussian measure; the model of a Gaussian random function; oscillations of sample functions; the convexity and isoperimetric inequalities; the regularity of sample functions of means of entropy characteristics and the majorizing measures; functional laws of the iterated logarithm and estimates for the probabilities of large deviations. This book should be of interest to mathematicians and scientists who use stochastic methods in their research and to students in probability theory.

This book investigates the distributions of functionals defined on the sample paths of stochastic processes. It contains systematic exposition and applications of three general research methods developed by the authors. The method of stratifications is used to study the problem of absolute continuity of distribution for different classes of functionals under very mild smoothness assumptions. It can be used also for evaluation of the distribution density of the functional. The method of differential operators is based on the abstract formalism of differential calculus and proves to be a powerful tool for the investigation of the smoothness properties of the distributions. The superstructure method, which is a later modification of the method of stratifications, is used to derive strong limit theorems (in the variation metric) for the distributions of stochastic functionals under weak convergence of the processes.Various application examples concern the functionals of Gaussian, Poisson and diffusion processes as well as partial sum processes from the Donsker-Prokhorov scheme. The research methods and basic results in this book are presented here in monograph form for the first time. The text would be suitable for a graduate course in the theory of stochastic processes and related topics.