Jaime San Martín - Böcker

This text contains the lectures given at the Conference on Dynamics and Randomness held at the Centro de Modelamiento Matematico of the Universidad de Chile, on December 11-15, 2000. This meeting brought together mathematicians, theoretical physicists, and theoretical computer scientists, and graduate students interested in fields related to probability theory, ergodic theory, and symbolic and topological dynamics. Each chapter is devoted to one of these subjects. Some papers are structured as surveys, presenting at the same time an original point of view and showing mostly new results.

This book contains the lectures given at the II Canference an Dynamics and Randamness held at the Centro de Modelamiento Matematico of the Universidad de Chile, from December 9th to 13th, 2002. This meeting brought together mathematicians, theoretical physicists, theoretical computer scientists, and graduate students interested in fields related to probability theory, ergodic theory, symbolic and topological dynamics. We would like to express our gratitude to an the participants of the conference and to the people who contributed to its orga- nization. In particular, to Pierre Collet, BerIiard Rost and Karl Petersen for their scientific advise.We want to thank warmly the authors of each chapter for their stimulating lectures and for their manuscripts devoted to a various of appealing subjects in probability and dynamics: to Jean Bertoin for his course on Some aspects of random fragmentation in con- tinuous time; to Anton Bovier for his course on Metastability and ageing in stochastic dynamics; to Steve Lalley for his course on AI- gebraic systems of generat ing functions and return probabilities for random walks; to Elon Lindenstrauss for his course on Recurrent measures and measure rigidity; to Sylvie Meleard for her course on Stochastic particle approximations for two-dimensional N avier- Stokes equations; and to Anatoly Vershik for his course on Random and universal metric spaces.

This book constitutes the refereed revised selected papers of the 11th Latin American Conference on High Performance Computing, CARLA 2024, held in Santiago de Chile, Chile, during September 30–October 4, 2024. The 21 full papers included in this book were carefully reviewed and selected from 42 submissions.

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.

The study of M-matrices, their inverses and discrete potential theory is now a well-established part of linear algebra and the theory of Markov chains. The main focus of this monograph is the so-called inverse M-matrix problem, which asks for a characterization of nonnegative matrices whose inverses are M-matrices. We present an answer in terms of discrete potential theory based on the Choquet-Deny Theorem. A distinguished subclass of inverse M-matrices is ultrametric matrices, which are important in applications such as taxonomy. Ultrametricity is revealed to be a relevant concept in linear algebra and discrete potential theory because of its relation with trees in graph theory and mean expected value matrices in probability theory. Remarkable properties of Hadamard functions and products for the class of inverse M-matrices are developed and probabilistic insights are provided throughout the monograph.

Main concepts of quasi-stationary distributions (QSDs) for killed processes are the focus of the present volume. For diffusions, the killing is at the boundary and for dynamical systems there is a trap. The authors present the QSDs as the ones that allow describing the long-term behavior conditioned to not being killed. Studies in this research area started with Kolmogorov and Yaglom and in the last few decades have received a great deal of attention. The authors provide the exponential distribution property of the killing time for QSDs, present the more general result on their existence and study the process of trajectories that survive forever. For birth-and-death chains and diffusions, the existence of a single or a continuum of QSDs is described. They study the convergence to the extremal QSD and give the classification of the survival process. In this monograph, the authors discuss Gibbs QSDs for symbolic systems and absolutely continuous QSDs for repellers. The findingsdescribed are relevant to researchers in the fields of Markov chains, diffusions, potential theory, dynamical systems, and in areas where extinction is a central concept. The theory is illustrated with numerous examples. The volume uniquely presents the distribution behavior of individuals who survive in a decaying population for a very long time. It also provides the background for applications in mathematical ecology, statistical physics, computer sciences, and economics.

Main concepts of quasi-stationary distributions (QSDs) for killed processes are the focus of the present volume. For diffusions, the killing is at the boundary and for dynamical systems there is a trap. The authors present the QSDs as the ones that allow describing the long-term behavior conditioned to not being killed. Studies in this research area started with Kolmogorov and Yaglom and in the last few decades have received a great deal of attention. The authors provide the exponential distribution property of the killing time for QSDs, present the more general result on their existence and study the process of trajectories that survive forever. For birth-and-death chains and diffusions, the existence of a single or a continuum of QSDs is described. They study the convergence to the extremal QSD and give the classification of the survival process. In this monograph, the authors discuss Gibbs QSDs for symbolic systems and absolutely continuous QSDs for repellers. The findingsdescribed are relevant to researchers in the fields of Markov chains, diffusions, potential theory, dynamical systems, and in areas where extinction is a central concept. The theory is illustrated with numerous examples. The volume uniquely presents the distribution behavior of individuals who survive in a decaying population for a very long time. It also provides the background for applications in mathematical ecology, statistical physics, computer sciences, and economics.

This volume is based on the lecture notes of six courses delivered at a CIMPA Summer School in Temuco, Chile, in January 2001. The courses are: asymptotic of the heat kernel in unbounded domains; spin systems with long range interactions; non-linear Dirichlet problem and non-linear integration; first-passage percolation; central limit theorem for Markov processes; stochastic orders and stopping times in Brownian motion. The level of each course is that of a graduate course, but the material will also be of interest for the specialist.

This book contains the lectures given at the II Canference an Dynamics and Randamness held at the Centro de Modelamiento Matematico of the Universidad de Chile, from December 9th to 13th, 2002. This meeting brought together mathematicians, theoretical physicists, theoretical computer scientists, and graduate students interested in fields related to probability theory, ergodic theory, symbolic and topological dynamics. We would like to express our gratitude to an the participants of the conference and to the people who contributed to its orga- nization. In particular, to Pierre Collet, BerIiard Rost and Karl Petersen for their scientific advise.We want to thank warmly the authors of each chapter for their stimulating lectures and for their manuscripts devoted to a various of appealing subjects in probability and dynamics: to Jean Bertoin for his course on Some aspects of random fragmentation in con- tinuous time; to Anton Bovier for his course on Metastability and ageing in stochastic dynamics; to Steve Lalley for his course on AI- gebraic systems of generat ing functions and return probabilities for random walks; to Elon Lindenstrauss for his course on Recurrent measures and measure rigidity; to Sylvie Meleard for her course on Stochastic particle approximations for two-dimensional N avier- Stokes equations; and to Anatoly Vershik for his course on Random and universal metric spaces.