From Classical to Modern Probability CIMPA Summer School 2001

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. This meeting brought together probabilists and graduate students interested in fields like particle systems, percolation, Brownian motion, random structures, potential theory and stochastic processes. We would like to express our gratitude to all the participants of the school as well as the people who contributed to its organization. In particular, to Servet Martinez, and Pablo Ferrari for their scientific advice, and Cesar Burgueiio for all his support and friendship. We want to thank all the professors for their stimulating courses and lectures. Special thanks to those who took the extra work in preparing each chapter of this book. We are also indebted to our sponsors and supporting institutions, whose interest and help was essential to organize this meeting: CIMPA, CNRS, CONI­ CYT, ECOS, FONDAP Program in Applied Mathematics, French Cooperation, Fundacion Andes, Presidential Fellowship, Universidad de Chile and Universidad de La Frontera. We are grateful to Miss Gladys Cavallone for her excellent work during the preparation of the meeting as well as for the considerable task of unifying the typography of the different chapters of this book.

Asymptotic of the Heat Kernel in Unbounded Domains.- 1 Introduction.- 2 Ratio theorem for domains with harmonic cone C? of dimension 1.- 3 Benedicks domains.- References.- Spin Systems with Long Range Interactions.- 1 Ising spin systems.- 2 Phase transition.- 3 Time evolution.- References.- Nonlinear Dirichlet Problem and Nonlinear Integration.- 1 Generalities.- 2 Preliminaries on capacities.- 3 Radon capacities.- 4 Lusin capacities.- 5 Answering the questions and last words.- References.- First-Passage Percolation.- 1 The basic model and the shape model.- 2 Exponential bounds for large deviations.- 3 Bounds of moderate deviations.- References.- Central Limit Theorem for Markov Processes.- 1 Introduction.- 2 A warming-up example.- 3 Central limit theorem for Markov processs.- 4 Regularity of the diffusion coefficient.- References.- Stochastic Orders and Stopping Times in Brownian Motion.- 1 Introduction.- 2 Martingales and stochastic orders.- 3 Skorohod embeddings in standard Brownian motion.- 4 A maximal property of Azéma—Yor stopped Brownian motion.- 5 Practical description of the optimal stopping time for the look-back American put option problem.- References.