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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.
Contents: Preface 1. Notations and Some Technical Results 2. Random Sets 3. Random Probability Measures and the Narrow Topology 4. Prohorov Theory for Random Probability Measures 5. Further Topologies on Random Measures 6. Invariant Measures and Some Ergodic theory for Random Dynamical Systems A. The Narrow Topology on Non-Random Measures B. Scattered Results Bibliography Index
Focusing on the mathematical description of stochastic dynamics in
discrete as well as in continuous time, this book investigates such
dynamical phenomena as perturbations, bifurcations and chaos. It
also introduces new ideas for the exploration of infinite
dimensional systems, in particular stochastic partial differential
equations. Example applications are presented from biology,
chemistry and engineering, while describing numerical treatments of
stochastic systems.
Focusing on the mathematical description of stochastic dynamics in
discrete as well as in continuous time, this book investigates such
dynamical phenomena as perturbations, bifurcations and chaos. It
also introduces new ideas for the exploration of infinite
dimensional systems, in particular stochastic partial differential
equations. Example applications are presented from biology,
chemistry and engineering, while describing numerical treatments of
stochastic systems.
Since the predecessor to this volume (LNM 1186, Eds. L. Arnold, V.
Wihstutz)appeared in 1986, significant progress has been made in
the theory and applications of Lyapunov exponents - one of the key
concepts of dynamical systems - and in particular, pronounced
shifts towards nonlinear and infinite-dimensional systems and
engineering applications are observable. This volume opens with an
introductory survey article (Arnold/Crauel) followed by 26 original
(fully refereed) research papers, some of which have in part survey
character. From the Contents: L. Arnold, H. Crauel: Random
Dynamical Systems.- I.Ya. Goldscheid: Lyapunov exponents and
asymptotic behaviour of the product of random matrices.- Y. Peres:
Analytic dependence of Lyapunov exponents on transition
probabilities.- O. Knill: The upper Lyapunov exponent of Sl (2, R)
cocycles: Discontinuity and the problem of positivity.- Yu.D.
Latushkin, A.M. Stepin: Linear skew-product flows and semigroups of
weighted composition operators.- P. Baxendale: Invariant measures
for nonlinear stochastic differential equations.- Y. Kifer: Large
deviationsfor random expanding maps.- P. Thieullen: Generalisation
du theoreme de Pesin pour l' -entropie.- S.T. Ariaratnam, W.-C.
Xie: Lyapunov exponents in stochastic structural mechanics.- F.
Colonius, W. Kliemann: Lyapunov exponents of control flows.
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