Normal Forms for Random Differential-Equations

Given a dynamical system (Ω, , , θ(t)) and a random differential equation = ƒ(θtω, x) in d with ƒ(ω, 0) = 0 a.s. The normal form problem is to construct a smooth near identity nonlinear random coordinate transformation h(ω) to make g(θt ω, y) := Dh(θt ω, y)−1 (ƒ(θt ω, h(θt ω, y)) − (d/dt) h(θt ω, y)) as simple as possible, preferably linear. The linearization Dƒ(θt, ω, 0) =: A(θt ω) generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eifenvalues (Lyapunov exponents) and eigenspnces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance of θt turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the concept of ε-normal form necessary. The stochastic versions of resonance and averaging are developed. One- and two-dimensional examples are treated in detail.