Normal forms for stochastic differential equations

Abstract

We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dx t m j0 f j x t d j t and dx t m j0 g j x t d j t in R d with smooth coecients satisfying f j 0 g j 0 0 are said to be smoothly equivalent if there is a smooth random dieomorphism (coordinate transformation) hx with hxY 0 0 and hhxY 0 id which conjugates the corresponding local¯ows,where h t xs xt s À xt is the (ergodic) shift on the canonical Wiener space. The normal form problem for SDE consists in ®nding the``simplest possible'' member in the equivalence class of a given SDE, in particular in giving conditions under which it can be linearized (g j x hf j 0x).We develop a mathematically rigorous normal form theory for SDE which justi®es the engineering and physics literature on that problem. It is based on the multiplicative ergodic theorem and uses a uniform (with respect to a spatial parameter) Stratonovich calculus which allows the handling of non-adapted initial values and coecients in the stochastic version of the cohomological equation. Our Probab. Theory Relat. Fields 110, 559±588 (1998) Key words and phrases: Stochastic dierential equation ± Stochastic normal form ± Stochastic cohomological equation ± Anticipative calculus ± Random dynamical system ± Multiplicative ergodic theory ± Ane stochastic dierential equation ± Noisy Dung-van der Pol oscillator. main result (Theorem 3.2) is that an SDE is (formally) equivalent to its linearization if the latter is nonresonant.As a by-product, we prove a general theorem on the existence of a stationary solution of an anticipative ane SDE.The study of the Dung-van der Pol oscillator with small noise concludes the paper.

Escape Rates in a Stochastic Environment with Multiple Scales

“…Rigorous, theoretical analysis to support normal form coordinate transforms (and center manifold reduction) was developed in [2,1]. In this work, the technical problem of the anticipative noise integrals also was dealt with rigorously.…”

“…Non-rigorous stochastic normal form analysis (which leads to the stochastic center manifold) was performed in [34,9,40,41]. Rigorous theoretical analysis of normal form coordinate transformations for stochastic center manifold reduction was developed in [2,1]. Later, an alternative method of stochastic normal form reduction was developed [42], in which any anticipatory convolutions (integrals into the future of the noise processes) that appeared in the slow modes were removed.…”

Invariant Manifolds for Stochastic Models in Fluid Dynamics

“…Just to name a few references, we will refer the reader to [1,2,14] and the bibliographies therein. Global invariant manifolds of stochastic evolution equations with linear one-dimensional noise have been studied in [8,9] under global Lipschitz nonlinearities and a spectral gap condition.…”

On Lie Group Classification of a Scalar Stochastic Differential Equation

“…In [11] there were introduced W -symmetries, which are fiber-preserving symmetries acting also on Wiener processes. It can be of interest to extend symmetry framework to very general transformations such as random diffeomorphisms of SDEs [3].…”