Hans Crauel - Böcker

In this monograph the narrow topology on random probability measures on Polish spaces is investigated in a thorough and comprehensive way. As a special feature, no additional assumptions on the probability space in the background, such as completeness or a countable generated algebra, are made. One of the main results is a direct proof of the random analog of the Prohorov theorem, which is obtained without invoking an embedding of the Polish space into a compact space. Further, the narrow topology is examined and other natural topologies on random measures are compared. In addition, it is shown that the topology of convergence in law-which relates to the "statistical equilibrium"-and the narrow topology are incompatible. A brief section on random sets on Polish spaces provides the fundamentals of this theory. In a final section, the results are applied to random dynamical systems to obtain existence results for invariant measures on compact random sets, as well as uniformity results in the individual ergodic theorem. This clear and incisive volume is useful for graduate students and researchers in mathematical analysis and its applications.

The conference on Random Dynamical Systems took place from April 28 to May 2, 1997, in Bremen and was organized by Matthias Gundlach and Wolfgang Kliemann with the help of th'itz Colonius and Hans Crauel. It brought together mathematicians and scientists for whom mathematics, in particular the field of random dynamical systems, is of relevance. The aim of the conference was to present the current state in the theory of random dynamical systems (RDS), its connections to other areas of mathematics, major fields of applications, and related numerical methods in a coherent way. It was, ho~vever, not by accident that the conference was centered around the 60th birthday of Ludwig Arnold. The theory of RDS o~ves much of its current state and status to Ludwig Arnold. Many aspects of the theory, a large number of results, and several substantial contributions were accomplished by Ludwig Arnold. An even larger number of contributions has been initiated by him. The field be- fited much from his enthusiasm, his openness for problems not completely aligned with his o~vn research interests, his ability to explain mathematics to researchers from other sciences as well as his ability to get mathema- clans interested in problems from applications not completely aligned with their research interests. In particular, a considerable part of the impact stochastics had on physical chemistry as well as on engineering goes back to Ludwig Arnold. He built up an active research group, kno~vn as "the Bremen group".

In this monograph the narrow topology on random probability measures on Polish spaces is investigated in a thorough and comprehensive way. As a special feature, no additional assumptions on the probability space in the background, such as completeness or a countable generated algebra, are made. One of the main results is a direct proof of the random analog of the Prohorov theorem, which is obtained without invoking an embedding of the Polish space into a compact space. Further, the narrow topology is examined and other natural topologies on random measures are compared. In addition, it is shown that the topology of convergence in law-which relates to the "statistical equilibrium"-and the narrow topology are incompatible. A brief section on random sets on Polish spaces provides the fundamentals of this theory. In a final section, the results are applied to random dynamical systems to obtain existence results for invariant measures on compact random sets, as well as uniformity results in the individual ergodic theorem. This clear and incisive volume is useful for graduate students and researchers in mathematical analysis and its applications.

The conference on Random Dynamical Systems took place from April 28 to May 2, 1997, in Bremen and was organized by Matthias Gundlach and Wolfgang Kliemann with the help of th'itz Colonius and Hans Crauel. It brought together mathematicians and scientists for whom mathematics, in particular the field of random dynamical systems, is of relevance. The aim of the conference was to present the current state in the theory of random dynamical systems (RDS), its connections to other areas of mathematics, major fields of applications, and related numerical methods in a coherent way. It was, ho~vever, not by accident that the conference was centered around the 60th birthday of Ludwig Arnold. The theory of RDS o~ves much of its current state and status to Ludwig Arnold. Many aspects of the theory, a large number of results, and several substantial contributions were accomplished by Ludwig Arnold. An even larger number of contributions has been initiated by him. The field be- fited much from his enthusiasm, his openness for problems not completely aligned with his o~vn research interests, his ability to explain mathematics to researchers from other sciences as well as his ability to get mathema- clans interested in problems from applications not completely aligned with their research interests. In particular, a considerable part of the impact stochastics had on physical chemistry as well as on engineering goes back to Ludwig Arnold. He built up an active research group, kno~vn as "the Bremen group".