Giuseppe da Prato Books

In this revised and extended version of his course notes from a 1-year course at Scuola Normale Superiore, Pisa, the author provides an introduction – for an audience knowing basic functional analysis and measure theory but not necessarily probability theory – to analysis in a separable Hilbert space of infinite dimension. Starting from the definition of Gaussian measures in Hilbert spaces, concepts such as the Cameron-Martin formula, Brownian motion and Wiener integral are introduced in a simple way. These concepts are then used to illustrate some basic stochastic dynamical systems (including dissipative nonlinearities) and Markov semi-groups, paying special attention to their long-time behavior: ergodicity, invariant measure. Here fundamental results like the theorems of Prokhorov, Von Neumann, Krylov-Bogoliubov and Khas'minski are proved. The last chapter is devoted to gradient systems and their asymptotic behavior.

The second edition features two new chapters that enhance the content, alongside an expanded bibliography for further reading. These updates aim to provide a more thorough understanding of the subject matter, making it an essential resource for readers seeking in-depth knowledge.

Focusing on second order linear parabolic and elliptic equations, this book presents a contemporary approach using probability measures in Hilbert and Banach spaces, along with stochastic evolution equations. It explores applications in various fields such as statistical mechanics, fluid mechanics, and control theory, highlighting the rapid development in these areas. The authors provide essential background material and numerous references for further study, making the book accessible to newcomers while offering advanced insights for experienced readers.

The book offers a comprehensive introduction to the analysis within a separable Hilbert space of infinite dimension, specifically tailored for readers familiar with basic functional analysis and measure theory, but not necessarily with probability theory. It builds on course notes from a year-long program at Scuola Normale Superiore in Pisa, providing both revised and extended content to enhance understanding of the subject.

This volume offers an introductory course on differential stochastic equations and Malliavin calculus, based on lectures at Scuola Normale Superiore di Pisa and other universities. It covers Gaussian measures, Brownian motion, Itô's formula, and applications like the Feynman-Kac formula. The third edition includes improvements and a new section on the Feynman-Kac semigroup.

Lectures: J. Chazarain, A. Piriou: Problèmes mixtes hyperboliques: Première partie: Les problèmes mixtes hyperboliques vérifiant 1a condition de Lopatinski uniforme; Deuxième partie: Propagation et réflexion des singularités.- L. Gårding: Introduction to hyperbolicity.- T. Kato: Linear and quasi-linear equations of evolution of hyperbolic type.- K.W. Morton: Numerical methods for non-linear hyperbolic equations of mathematical physics.- Seminars: H. Brezis: First-order quasilinear equation on a torus.

This book consists of five introductory contributions by leading mathematicians on the functional analytic treatment of evolutions equations. In particular the contributions deal with Markov semigroups, maximal L^p-regularity, optimal control problems for boundary and point control systems, parabolic moving boundary problems and parabolic nonautonomous evolution equations. The book is addressed to PhD students, young researchers and mathematicians doing research in one of the above topics.

This textbook gives an introduction to stochastic partial differential equations such as reaction-diffusion, Burgers and 2D Navier-Stokes equations, perturbed by noise. Several properties of corresponding transition semigroups are studied, such as Feller and strong Feller properties, irreducibility, existence and uniqueness of invariantg measures. Moreover, the transition semigroups are interpreted as generalized solutions of Kologorov equations. The prerequisites are basic probability (including finite dimemsional stochastic differential equations), basic functional analysis and some elements of the theory of partial differential equations.