Summary.
We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dx t = ∑ j =0 m f j (x t )∘dW t j and dx t =∑ j =0 m g j (x t )∘dW t j in ℝd with smooth coefficients satisfying f j (0)=g j (0)=0 are said to be smoothly equivalent if there is a smooth random diffeomorphism (coordinate transformation) h(ω) with h(ω,0)=0 and Dh(ω,0)=id which conjugates the corresponding local flows,
where θ t ω(s)=ω(t+s)−ω(t) is the (ergodic) shift on the canonical Wiener space. The normal form problem for SDE consists in finding 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)=Df j (0)x).
We develop a mathematically rigorous normal form theory for SDE which justifies 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 coefficients in the stochastic version of the cohomological equation. Our 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 affine SDE.
The study of the Duffing-van der Pol oscillator with small noise concludes the paper.
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Received: 19 August 1997 / In revised form: 15 December 1997
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Arnold, L., Imkeller, P. Normal forms for stochastic differential equations. Probab Theory Relat Fields 110, 559–588 (1998). https://doi.org/10.1007/s004400050159
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DOI: https://doi.org/10.1007/s004400050159