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Normal Forms for Random Differential-Equations by L. Arnold; K.D. Xu is a Mathematics article available to read on EtoBox.

What is Normal Forms for Random Differential-Equations about?

Given a dynamical system \(\left(\Omega, \mathscr{F}, P, Q_{1}\right)\) and a random differential equation \(\dot{x}=f\left(\theta,(\omega, x)\right.\) in \(\mathbb{R}^{d}\) with \(f(\omega, 0)=0\) a.s. The normal form problem is to construct a smooth near identity nonlinear random coordinate transformation \(h(\omega)\) to make \(g(\theta,(\omega, y):=D h(\theta,(\omega, y) \cdot(f i(\theta, \omega, h(\theta, \omega, y))-(d / d) h(\theta,(\omega, y))\) as simple as possible, preferably linear. The linearization \(D f\left(\theta_{t}(\omega, 0)=: A(\theta, \omega)\right.\) generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one. the difference being that the appearance of \(\theta\), turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the concept of \(\varepsilon\)-normal form neeessary. The stochastic versions of resonance and averaging are developed. One- and (wo-dimensional examples are treated in detail. ' 1995

Who reads Normal Forms for Random Differential-Equations?

Author

L. Arnold; K.D. Xu

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0022-0396)

Published

1995

Language

EN

Field

Mathematics (Physical Sciences)