Normal Forms for Stochastic Diierential Equations

We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dxt = P m j=0 fj (xt) dW j t and dxt = P m j=0 gj(xt)dW j t in R d with smooth coeecients satisfying fj (0) = gj(0) = 0 are said to be smoothly equivalent if there is a smooth random diieomorphism (coordinate transformation) h(!) with h(!; 0) = 0 and Dh(!; 0) = id which conjugates the corresponding local ows, '(t; !) h(!) = h(t!) (t; !); where t!(s) = !(t + s) ? !(t) 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 (gj(x) = Dfj (0)x). We develop a mathematically rigorous normal form theory for SDE which justiies 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 coeecients in the stochastic version of the cohomological equation. As a by{product, we prove a general theorem on the existence of a stationary solution of an anticipative aane SDE. 1 The study of the Duung{van der Pol oscillator with small noise concludes the paper.