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Stochastic Nonlinear Systems. Existing Methods and New Results
Author(s) -
Sobczyk K.
Publication year - 1998
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/(sici)1521-4001(199810)78:10<651::aid-zamm651>3.0.co;2-s
Subject(s) - nonlinear system , mathematics , continuous time stochastic process , stochastic calculus , stochastic process , brownian motion , uniqueness , principle of maximum entropy , stochastic differential equation , stochastic partial differential equation , mathematical analysis , statistical physics , partial differential equation , physics , statistics , quantum mechanics
In this article the most essential features and recent problems of stochastic nonlinear dynamics are presented with an emphasis on new methods and results of special significance in applications. Starting from the historical origin of stochastic dynamics (the Langevin equation for a Brownian particle), the general governing nonlinear equations for a wide class of real systems are presented along with their possible interpretations (e.g. regular stochastic systems, Itô and Stratonovich equations, stochastic equations in Hilbert space). The most notable qualitative and quantitative problems and methods for stochastic nonlinear systems are expounded and illustrated; for example: the existence and uniqueness theorem and its restrictions, explosions of the solutions, Khasminskii limit theorem (stochastic averaging method), stationary solutions, noise‐induced phase transitions, and stochastic bifurcations. Amongst the quantitative methods the equations for the statistical moments of the solution process are presented and then used to construct the probability distributions via the maximum entropy method.