In November 2004, M. Yor and R. Mansuy jointly gave six lectures at Columbia University, New York. These notes follow the contents of that course, covering expansion of filtration formulae; BDG inequalities up to any random time; martingales that vanish on the zero set of Brownian motion; the Azema-Emery martingales and chaos representation; the filtration of truncated Brownian motion; attempts to characterize the Brownian filtration.

The book accordingly sets out to acquaint its readers with the theory and main examples of enlargements of filtrations, of either the initial or the...

This book demonstrates that while elliptic and hyperbolic tori determine the distribution of maximal invariant tori, they themselves form n-parameter families. Therefore, torus bifurcations of high co-dimension may be found in a single given Hamiltonian system, absent untypical conditions or external parameters. The text moves logically from the integrable case, in which symmetries allow for reduction to bifurcating equilibria, to non-integrability, where smooth parametrisations must be replaced by Cantor sets.

This book teams up the spectral theory of bounded linear operators with von Neumann's theory of unbounded operators to provide a framework for the study of stable methods for the evaluation of unbounded operators. The text presents numerous illustrations of unbounded linear operators that arise in various inverse problems of mathematical physics. It also offers an extensive exposition of background material from the theory of operators on Hilbert space.

The territory of preserver problems has grown continuously within linear analysis. This book presents a cross-section of the modern theory of preservers on infinite dimensional spaces (operator spaces and function spaces) through the author's corresponding results. Special emphasis is placed on preserver problems concerning some structures of Hilbert space operators which appear in quantum mechanics. In addition, local automorphisms and local isometries of operator algebras and function algebras are discussed in detail.

This book serves as a concise introduction to the state-of-the-art of spin glass theory. The collection of review papers are written by leading experts in the field and cover the topic from a wide variety of angles. The book will be useful to both graduate students and young researchers, as well as to anyone curious to know what is going on in this exciting area of mathematical physics.

This expanded version of the 1997 European Mathematical Society Lectures given by the author in Helsinki, begins with a self-contained introduction to nonstandard analysis (NSA) and the construction of Loeb Measures, which are rich measures discovered in 1975 by Peter Loeb, using techniques from NSA. Subsequent chapters sketch a range of recent applications of Loeb measures due to the author and his collaborators, in such diverse fields as (stochastic) fluid mechanics, stochastic calculus of variations ("Malliavin" calculus) and the mathematical finance theory. The exposition is designed for...

This is the first book to present a model, based on rational mechanics of electrorheological fluids, that takes into account the complex interactions between the electromagnetic fields and the moving liquid. Several constitutive relations for the Cauchy stress tensor are discussed. The main part of the book is devoted to a mathematical investigation of a model possessing shear-dependent viscosities, proving the existence and uniqueness of weak and strong solutions for the steady and the unsteady case. The PDS systems investigated possess so-called non-standard growth conditions. Existence...

Since the time of surfaces -+ in differential Gauss, parametrized (x, y) P(x, y) have been described a frame attached to the moving geometry through TI(x, y) surface. One introduces the Gauss- which linear dif- Weingarten equations are, ferential equations = U = TIX T1, VT', PY (1. for the and their condition frame, compatibility - = V + U, V] 0, UY (1.2) which the Gauss-Codazzi For surfaces in three-dim- represents equations . a sional Euclidean the frame T1 lies in the usually or space, group SO(3) SU(2). On the other a of a non-linear in the form hand, representation equation (1.2) is...

The main objective of this monograph is the study of a class of stochastic differential systems having unbounded coefficients, both in finite and in infinite dimension. We focus our attention on the regularity properties of the solutions and hence on the smoothing effect of the corresponding transition semigroups in the space of bounded and uniformly continuous functions. As an application of these results, we study the associated Kolmogorov equations, the large-time behaviour of the solutions and some stochastic optimal control problems together with the corresponding Hamilton-...

The notion of amenability has its origins in the beginnings of modern measure theory: Does a finitely additive set function exist which is invariant under a certain group action? Since the 1940s, amenability has become an important concept in abstract harmonic analysis (or rather, more generally, in the theory of semitopological semigroups). In 1972, B.E. Johnson showed that the amenability of a locally compact group G can be characterized in terms of the Hochschild cohomology of its group algebra L DEGREES1(G): this initiated the theory of amenable Banach algebras. Since then, amenability...