Dynamics Of Stochastic Systems
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Dynamics of Stochastic Systems
Fluctuating parameters appear in a variety of physical systems and phenomena. They typically come either as random forces/sources, or advecting velocities, or media (material) parameters, like refraction index, conductivity, diffusivity, etc. The well known example of Brownian particle suspended in fluid and subjected to random molecular bombardment laid the foundation for modern stochastic calculus and statistical physics. Other important examples include turbulent transport and diffusion of particle-tracers (pollutants), or continuous densities (''oil slicks''), wave propagation and scattering in randomly inhomogeneous media, for instance light or sound propagating in the turbulent atmosphere.Such models naturally render to statistical description, where the input parameters and solutions are expressed by random processes and fields.The fundamental problem of stochastic dynamics is to identify the essential characteristics of system (its state and evolution), and relate those to the input parameters of the system and initial data.This raises a host of challenging mathematical issues. One could rarely solve such systems exactly (or approximately) in a closed analytic form, and their solutions depend in a complicated implicit manner on the initial-boundary data, forcing and system's (media) parameters . In mathematical terms such solution becomes a complicated "nonlinear functional" of random fields and processes.Part I gives mathematical formulation for the basic physical models of transport, diffusion, propagation and develops some analytic tools.Part II sets up and applies the techniques of variational calculus and stochastic analysis, like Fokker-Plank equation to those models, to produce exact or approximate solutions, or in worst case numeric procedures. The exposition is motivated and demonstrated with numerous examples.Part III takes up issues for the coherent phenomena in stochastic dynamical systems, described by ordinary and partial differential equations, like wave propagation in randomly layered media (localization), turbulent advection of passive tracers (clustering).Each chapter is appended with problems the reader to solve by himself (herself), which will be a good training for independent investigations.·This book is translation from Russian and is completed with new principal results of recent research.·The book develops mathematical tools of stochastic analysis, and applies them to a wide range of physical models of particles, fluids, and waves.·Accessible to a broad audience with general background in mathematical physics, but no special expertise in stochastic analysis, wave propagation or turbulence
Recent Development in Stochastic Dynamics and Stochastic Analysis
1. Hyperbolic equations with random boundary conditions / Zdzisław Brzeźniak and Szymon Peszat -- 2. Decoherent information of quantum operations / Xuelian Cao, Nan Li and Shunlong Luo -- 3. Stabilization of evolution equations by noise / Tomás Caraballo and Peter E. Kloeden -- 4. Stochastic quantification of missing mechanisms in dynamical systems / Baohua Chen and Jinqiao Duan -- 5. Banach space-valued functionals of white noise / Yin Chen and Caishi Wang -- 6. Hurst index estimation for self-similar processes with long-memory / Alexandra Chronopoulou and Frederi G. Viens -- 7. Modeling colored noise by fractional Brownian motion / Jinqiao Duan, Chujin Li and Xiangjun Wang -- 8. A sufficient condition for non-explosion for a class of stochastic partial differential equations / Hongbo Fu, Daomin Cao and Jinqiao Duan -- 9. The influence of transaction costs on optimal control for an insurance company with a new value function / Lin He, Zongxia Liang and Fei Xing -- 10. Limit theorems for p-variations of solutions of SDEs driven by additive stable Lévy noise and model selection for paleo-climatic data / Claudia Hein, Peter Imkeller and Ilya Pavlyukevich -- 11. Class II semi-subgroups of the infinite dimensional rotation group and associated Lie algebra / Takeyuki Hida and Si Si -- 12. Stopping Weyl processes / Robin L. Hudson -- 13. Karhunen-Loéve expansion for stochastic convolution of cylindrical fractional Brownian motions / Zongxia Liang -- 14. Stein's method meets Malliavin calculus : a short survey with new estimates / Ivan Nourdin and Giovanni Peccati -- 15. On stochastic integrals with respect to an infinite number of Poisson point process and its applications / Guanglin Rang, Qing Li and Sheng You -- 16. Lévy white noise, elliptic SPDEs and Euclidean random fields / Jiang-Lun Wu -- 17. A short presentation of Choquet integral / Jia-An Yan
Stochastic Dynamics of Structures
In Stochastic Dynamics of Structures, Li and Chen present a unified view of the theory and techniques for stochastic dynamics analysis, prediction of reliability, and system control of structures within the innovative theoretical framework of physical stochastic systems. The authors outline the fundamental concepts of random variables, stochastic process and random field, and orthogonal expansion of random functions. Readers will gain insight into core concepts such as stochastic process models for typical dynamic excitations of structures, stochastic finite element, and random vibration analysis. Li and Chen also cover advanced topics, including the theory of and elaborate numerical methods for probability density evolution analysis of stochastic dynamical systems, reliability-based design, and performance control of structures. Stochastic Dynamics of Structures presents techniques for researchers and graduate students in a wide variety of engineering fields: civil engineering, mechanical engineering, aerospace and aeronautics, marine and offshore engineering, ship engineering, and applied mechanics. Practicing engineers will benefit from the concise review of random vibration theory and the new methods introduced in the later chapters. "The book is a valuable contribution to the continuing development of the field of stochastic structural dynamics, including the recent discoveries and developments by the authors of the probability density evolution method (PDEM) and its applications to the assessment of the dynamic reliability and control of complex structures through the equivalent extreme-value distribution." —A. H-S. Ang, NAE, Hon. Mem. ASCE, Research Professor, University of California, Irvine, USA "The authors have made a concerted effort to present a responsible and even holistic account of modern stochastic dynamics. Beyond the traditional concepts, they also discuss theoretical tools of recent currency such as the Karhunen-Loeve expansion, evolutionary power spectra, etc. The theoretical developments are properly supplemented by examples from earthquake, wind, and ocean engineering. The book is integrated by also comprising several useful appendices, and an exhaustive list of references; it will be an indispensable tool for students, researchers, and practitioners endeavoring in its thematic field." —Pol Spanos, NAE, Ryon Chair in Engineering, Rice University, Houston, USA