Stochastic Equations for Complex Systems

ISBN-13: 9789401081771 / Angielski / Miękka / 2012 / 175 str.

A. V. Skorohod

Opis

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the tinal question. G. K. Chesterton. The Scandal oj Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics, However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches, It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related, Further. the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non- trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces, And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes, They draw upon widely different sections of mathematics.

Spis treści

1. Continuous Markov Processes in a Locally Compact Space and Stochastic Equations.- 1. Quasi-diffusion Processes.- 2. Stochastic Equations for Quasi-diffusion Processes.- 3. Existence and Uniqueness of the Solution of a Stochastic Differential Equation. The Smooth Case.- 4. Limit Theorems for Solutions of Stochastic Equations.- 5. Weak Solutions.- 6. Stochastic Equations in Rm.- 2. Randomly Interacting Systems of Particles.- 1. Stochastic Equation for Systems of Randomly Interacting Particles.- 2. Problem of the Asymptotic Behavior of a Statistical Distribution Function. Compactness Conditions.- 3. Limit Theorem for Steplike Processes.- 4. Limit Theorem for Statistical Distribution Functions. General Case.- 5. Fluctuations.- 6. Limit Theorem for Functionals.- 7. Diffusion Approximation.