Imkeller, Peter; Schmalfuss, Björn The conjugacy of stochastic and random differential equations and the existence of global attractors. (English) Zbl 1004.37034 J. Dyn. Differ. Equations 13, No. 2, 215-249 (2001). The paper studies stochastic differential equations of the form \[ dx=f_0(x) dt+\sum_{k=1}^mf_k(x) dW_k(t)\tag{1} \] on \(\mathbb R^d\) with \(C^\infty\) vector fields \(f_k\), \(0\leq k\leq m\), such that the \(f_k\) are globally Lipschitz and commute for \(1\leq k\leq m\) (i. e., \([f_k,f_n]=0\) for \(1\leq k,n\leq m\), where \([\cdot,\cdot]\) denotes the Lie bracket). Assuming the stochastic flow induced by (1) to be forward complete, a random stationary co-ordinate transformation is constructed, such that the stochastic flow induced by (1) is conjugate to the flow induced by a certain random differential equation. This coordinate transformation transforms attractors for (1) to attractors for the random differential equation. This method, which had been used previously several times in the literature in a more ad hoc fashion, is then used to prove the existence of random attractors for several examples of the form (1) (viz the Duffing-van der Pol oscillator with multiplicative noise on the position, on the velocity, and with additional additive noise, resp., the noisy damped harmonic oscillator in a double-well potential, and the Lorenz system with full and with partial multiplicative noise). Reviewer: Hans Crauel (Ilmenau) Cited in 2 ReviewsCited in 65 Documents MSC: 37H10 Generation, random and stochastic difference and differential equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34F05 Ordinary differential equations and systems with randomness 37H99 Random dynamical systems 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems 37C60 Nonautonomous smooth dynamical systems Keywords:random coordinate transformation; conjugacy of stochastic and random differential equation; Lyapunov function; random attractors; stochastic Duffing-van der Pol oscillator; stochastic Lorenz system PDFBibTeX XMLCite \textit{P. Imkeller} and \textit{B. Schmalfuss}, J. Dyn. Differ. Equations 13, No. 2, 215--249 (2001; Zbl 1004.37034) Full Text: DOI