Stochastics

In 1931 Erwin Schrodinger considered the following problem: A huge cloud of independent and identical particles with known dynamics is supposed to be observed at finite initial and final times.

The following notes represent approximately the second half of the lectures I gave in the Nachdiplomvorlesung, in ETH, Zurich, between October 1991 and February 1992, together with the contents of six additional lectures I gave in ETH, in November and December 1993.

Disordered systems are statistical mechanics models in random environments.

What is high dimensional probability?

Harmonic analysis and probability have long enjoyed a mutually beneficial relationship that has been rich and fruitful.

Discrete stochastics is the theory of discrete probability spaces.

"e;Elegantly written, with obvious appreciation for fine points of higher mathematics.

This book contains the lecture notes for a DMV course presented by the authors at Gunzburg, Germany, in September, 1990.

Schrodinger Equations and Diffusion Theory addresses the question "e;What is the Schrodinger equation?

Statistics is strongly tied to applications in different scientific disciplines, and the most challenging statistical problems arise from problems in the sciences.

The proceedings of the summer 1999 Chorin workshop on stochastic climate models captures well the spirit of enthusiasm of the workshop participants engaged in research in this exciting field.

This volume contains a selection of invited papers, presented to the fourth In- Statistical Analysis Based on the L1-Norm and Related ternational Conference on Methods, held in Neuchatel, Switzerland, from August 4-9, 2002.

Commonly there is no natural place in a traditional curriculum for mathematics or statistics, where a bridge between theory and practice fits into.

There are two parts in this book.

The seventh International Conference on Evolution Equations and their main areas of Applications (where the emphasis evolves as time and problems change) was held October 30 to November 4 at the CIRM (Centro Internazionale per la Ricerca Matematica) in Trento, Italy.

Concentration inequalities, which express the fact that certain complicated random variables are almost constant, have proven of utmost importance in many areas of probability and statistics.

The title High Dimensional Probability is an attempt to describe the many trib- utaries of research on Gaussian processes and probability in Banach spaces that started in the early 1970's.

This volume is based on lectures notes for the courses delivered at the Cimpa Summer School: From Classical to Modern Probability, held at Temuco, Chile, be- th th tween January 8 and 26 , 2001.

On what grounds can one reasonably expect that a complex financial contract solving a complex real-world issue does not deserve the same thorough scientific treatment as an aeroplane wing or a micro-proces- sor?

During the of Fall 1991, The Centre de Recerca Matematica, a research institute sponsored by the Institut d'Estudis Catalans, devoted a quarter to the study of stochastic analysis.

The last decades have demonstrated that quantum mechanics is an inexhaustible source of inspiration for contemporary mathematical physics.

A beautiful interplay between probability theory (Markov processes, martingale theory) on the one hand and operator and spectral theory on the other yields a uniform treatment of several kinds of Hamiltonians such as the Laplace operator, relativistic Hamiltonian, Laplace-Beltrami operator, and generators of Ornstein-Uhlenbeck processes.

"e;Stochastic Processes in Quantum Physics"e; addresses the question 'What is the mathematics needed for describing the movement of quantum particles', and shows that it is the theory of stochastic (in particular Markov) processes and that a relativistic quantum particle has pure-jump sample paths while sample paths of a non-relativistic quantum particle are continuous.

The second conference on Fractal Geometry and Stochastics was held at Greifs- wald/Koserow, Germany from August 28 to September 2, 1998.

This book is about discrete-time, time-homogeneous, Markov chains (Mes) and their ergodic behavior.

The seminar on Stochastic Analysis and Mathematical Physics of the Ca- tholic University of Chile, started in Santiago in 1984, has being followed and enlarged since 1995 by a series of international workshops aimed at pro- moting a wide-spectrum dialogue between experts on the fields of classical and quantum stochastic analysis, mathematical physics, and physics.

This book contains the proceedings of the conference "e;Fractals in Graz 2001 - Analysis, Dynamics, Geometry, Stochastics"e; that was held in the second week of June 2001 at Graz University of Technology, in the capital of Styria, southeastern province of Austria.

These notes had their origin in a postgraduate lecture series I gave at the Eid- genossiche Technische Hochschule (ETH) in Zurich in the Spring of 2000.

This volume contains the Proceedings of the Fourth Seminar on Stochastic Analy- sis, Random Fields and Applications, which took place at the Centro Stefano Fran- scini (Monte Verita) in Ascona (Ticino), Switzerland, from May 20 to 24, 2002.

Fractal geometry is used to model complicated natural and technical phenomena in various disciplines like physics, biology, finance, and medicine.

Since the publication of the first edition of this seminar book in 1994, the theory and applications of extremes and rare events have enjoyed an enormous and still increasing interest.

According to Leo Breiman (1968), probability theory has a right and a left hand.

Fractal geometry is a new and promising field for researchers from different disciplines such as mathematics, physics, chemistry, biology and medicine.

Pure and applied stochastic analysis and random fields form the subject of this book.

Robust Control is one of the fastest growing and promising areas of research today.

The statistical analysis of extreme data is important for various disciplines, including hydrology, insurance, finance, engineering and environmental sciences.

This book presents in thirteen refereed survey articles an overview of modern activity in stochastic analysis, written by leading international experts.

This thirteenth volume of the Poincare Seminar Series, Henri Poincare, 1912-2012, is published on the occasion of the centennial of the death of Henri Poincare in 1912.

This volume presents self-contained survey articles on modern research areas written by experts in their fields.

An Introduction to Quantum Stochastic Calculus aims to deepen our understanding of the dynamics of systems subject to the laws of chance both from the classical and the quantum points of view and stimulate further research in their unification.

Schrodinger Equations and Diffusion Theory addresses the question "e;What is the Schrodinger equation?

This volume contains refereed research or review articles presented at the 7th Seminar on Stochastic Analysis, Random Fields and Applications which took place at the Centro Stefano Franscini (Monte Verita) in Ascona , Switzerland, in May 2011.

This is a collection of papers by participants at High Dimensional Probability VI Meeting held from October 9-14, 2011 at the Banff International Research Station in Banff, Alberta, Canada.